Development of Bayesian geostatistical models with ... · malaria epidemiology INAUGURALDISSERTATION zur ... Bayesian computation implemented via Markov chain Monte Carlo (MCMC) simulation

Development of Bayesiangeostatistical modelswith applications inmalaria epidemiology

INAUGURALDISSERTATIONzur

Erlangung der Wurde eines Doktors der Philosophie

vorgelegt derPhilosophisch-Naturwissenschaftlichen Fakultat

der Universitat Basel

vonLaura Gosoniu

aus Bukarest, Rumanien

Basel, December 2008

Genehmigt von der Philosophisch-Naturwissenschaftlichen Fakultat auf Antrag von Prof.

Dr. M. Tanner, Dr. P. Vounatsou, Prof. Dr. T. Smith und Prof. Dr. M. Schumacher.

Basel, den 19. Dezember 2007

Prof. Dr. Hans-Peter Hauri

Dekan

Dedicated to my parents

and my beloved husband, Dominic

iv

Summary

Plasmodium falciparum malaria is a leading infectious disease and a major cause of mor-

bidity and mortality in large areas of the developing world, especially Africa. Accurate

estimates of the burden of the disease are useful for planning and implementing malaria

control interventions and for monitoring the impact of prevention and control activities.

Information on the population at risk of malaria can be compared to existing levels of

service provision to identify underserved populations and to target interventions to high

priority areas. The current available statistics for malaria burden are not reliable because

of the poor malaria case reporting systems in most African countries and the lack of na-

tional representative malaria surveys. Accurate maps of malaria distribution together with

human population totals are valuable tools for generating valid estimates of population at

risk.

Empirical mapping of the geographical patterns of malaria transmission in Africa requires

field survey data on prevalence of infection. The Mapping Malaria Risk in Africa (MARA)

is the most comprehensive database on malariological survey data across all sub-Sahara

Africa. Transmission of malaria is environmentally driven because it depends on the dis-

tribution and abundance of mosquitoes, which are sensitive to environmental and climatic

conditions. Estimating the environment-disease relation, the burden of malaria can be

predicted at places where data on transmission are not available. Malaria data collected

at fixed locations over a continuous study area (geostatistical data) are correlated in space

because common exposures of the disease influence malaria transmission similarly in neigh-

boring areas. Geostatistical models take into account spatial correlation by introducing

location-specific random effects. Geographical dependence is considered as a function of

the distance between locations. These models are highly parametrized. State-of-the-art

Bayesian computation implemented via Markov chain Monte Carlo (MCMC) simulation

methods enables model fit. A common assumption in geostatistical modeling of malaria

v

data is the stationarity, that is the spatial correlation is a function of distance between

locations and not of the locations themselves. This hypothesis does not always hold, es-

pecially when modeling malaria over large areas, hence geostatistical models that take

into account non-stationarity need to be assessed. Fitting geostatistical models requires

repeated inversions of the variance-covariance matrix modeling geographical dependence.

For very large number of data locations matrix inversion is considered infeasible. Methods

for optimizing this computation are needed. In addition, the relation between environmen-

tal factors and malaria risk is often not linear and parametric functions may not be able

to determine the shape of the relationship. Nonparametric geostatistical regression models

that allow the data to determine the form of the environment-malaria relation need to be

further developed and applied in malaria mapping.

The aim of this thesis was to develop appropriate models for non-stationary and large geo-

statistical data that can be applied in the field of malaria epidemiology to produce accurate

maps of malaria distribution. The main contributions of this thesis are the development

of methods for: (i) analyzing non-stationary malaria survey data; (ii) modeling the non-

linear relation between malaria risk and environment/climatic conditions; (iii) modeling

geostatistical mortality data collected at very large number of locations and (iv) adjusting

for seasonality and age in mapping heterogeneous malaria survey data.

Chapter 2 assessed the spatial effect of bednets on all-cause child mortality by analyzing

data from a large follow-up study in an area of high perennial malaria transmission in

Kilombero Valley, southern Tanzania. The results indicated a lack of community effect

of bednets density possibly because of the homogeneous characteristic of nets coverage

and the small proportion of re-treated nets in the study area. The mortality data of this

application were collected over 7, 403 locations. To overcome large matrix inversion a

Bayesian geostatistical model was developed. This model estimates the spatial process by

a subset of locations and approximates the location-specific random effects by a weighted

sum of the subset of location-specific random effects with the weights inversely proportional

to the separation distance.

In Chapter 3 a Bayesian non-stationary model was developed by partitioning the study

region into fixed subregions, assuming a separate stationary spatial process in each tile and

taking into account between-tile correlation. This methodology was applied on malaria

survey data extracted from the MARA database and produced parasitaemia risk maps in

Mali. The predictive ability of the non-stationary model was compared with the stationary

vi

analogue and the results showed that the stationarity assumption influenced the significance

of environmental predictors as well as the the estimation of the spatial parameters. This

indicates that the assumptions about the spatial process play an important role in inference.

Model validation showed that the non-stationary model had better predictive ability. In

addition, experts opinion suggested that the parasitaemia risk map based on the non-

stationary model reflects better the malaria situation in Mali. This work revealed that

non-stationarity is an essential characteristic which should be considered when mapping

malaria.

Chapter 4 employed the above non-stationary model to produce maps of malaria risk in

West Africa considering as fixed tiles the four agro-ecological zones that partition the re-

gion. Non-linearity in the relation between parasitaemia risk and environmental conditions

was assessed and it was addressed via P-splines within a Bayesian geostatistical model for-

mulation. The model allowed a separate malaria-environment relation in each zone. The

discontinuities at the borders between the zones were avoided since the spatial correlation

was modeled by a mixture of spatial processes over the entire study area, with the weights

chosen to be exponential functions of the distance between the locations and the centers

of the zones corresponding to each of the spatial processes.

The above modeling approach is suitable for mapping malaria over areas with an obvious

fixed partitioning (i.e. ecological zones). For areas where this is not possible, a non-

stationary model was developed in Chapter 5 by allowing the data to decide on the number

and shape of the tiles and thus to determine the different spatial processes. The partitioning

of the study area was based on random Voronoi tessellations and model parameters were

estimated via reversible jump Markov chain Monte Carlo (RJMCMC) due to the variable

dimension of the parameter space.

In Chapter 6 the feasibility of using the recently developed mathematical malaria trans-

mission models to adjust for age and seasonality in mapping historical malaria survey data

was investigated. In particular, the transmission model was employed to translate age

heterogeneous survey data from Mali into a common measure of transmission intensity. A

Bayesian geostatistical model was fitted on the transmission intensity estimates using as

covariates a number of environmental/climatic variables. Bayesian kriging was employed

to produce smooth maps of transmission intensity, which were further converted to age

specific parasitaemia risk maps. Model validation on a number of test locations showed

that this transmission model gives better predictions than modeling directly the prevalence

vii

data. This approach was further validated by analyzing the nationally representative ma-

laria surveys data derived from the Malaria Indicator surveys (MIS) in Zambia. Although

MIS data do not have the same limitations with the historical data, the purpose of the an-

alyzes was to compare the maps obtained by modeling 1) directly the raw prevalence data

and 2) transmission intensity data derived via the transmission model. Both maps pre-

dicted similar patterns of malaria risk, however the map based on the transmission model

predicted a slightly higher lever of endemicity. The use of transmission models on malaria

mapping enables adjusting for seasonality and age dependence of malaria prevalence and

it includes all available historical data collected at different age groups.

viii

Zusammenfassung

Plasmodium falciparum Malaria ist eine der haufigsten infektiosen Krankheiten und der

Hauptverursacher von Morbiditat und Sterblichkeit in weiten Teilen der Dritten Welt,

besonders in Afrika. Genaue Abschatzungen zur Belastung durch die Erkrankung sind hil-

freich fur die Planung und Durchfuhrung von Malaria Interventionen und fur die uberwachung

von Praventions- und Kontrolleinflussen. Informationen uber die gefahrdeten Bevolkerungen

durch Malaria konnen mit verschiedenen existierenden Modellen verglichen werden um be-

nachteiligte Gruppen zu identifizieren und gezielt Verbeugungen in den wichtigsten Gebi-

eten zu ergreifen. Die derzeitig verfugbaren statistischen Methoden zur Bestimmung der

Belastung durch Malaria sind allerdings nicht sehr zuverlassig, da in den meisten afrikanis-

chen Landern nur ein durftiges Meldesystem fur Malariafalle besteht. Ausserdem fehlen

landesweite reprasentative Studien. Korrekte Karten zur Malariatransmission zusammen

mit den Gesamtzahlen der Bevolkerung sind nutzliche Werkzeuge um Abschatzungen uber

die gefahrdete Bevolkerung zu erhalten.

Empirische Karten uber die geografischen Muster der Malaria Verbreitung in Afrika benotigen

Pravalenzdaten aus Studien uber die Erkrankung. Die Datenbank ”Mapping Malaria Risk

in Africa (MARA)” ist die umfangreichste ihrer Art welche Daten uber Malaria bezogene

Studien in Afrika sudlich der Sahara sammelt. Die Ausbreitung von Malaria wird durch

okologische Faktoren beeinflusst, weil die Erkrankung von der Verteilung und Menge von

Moskitos abhangig ist, welche empfindsam auf Umwelt und Klima reagieren. Durch die

Einbeziehung der Korrelation von Umwelt und Erkrankung kann die Belastung durch Ma-

laria selbst an Platzen abgeschatzt werden uber die ansonsten keine weiteren Daten zur

Verbreitung von Malaria zur Verfugung stehen. Daten die an einer bestimmten Anzahl von

Orten gesammelt wurden (geostatistische Daten) sind raumlich korreliert, da die bekan-

nten Einflussfaktoren die Malariatransmission zueinander ahnlich in benachbarten Gebi-

eten beeinflussen. Geostatistische Modelle berucksichtigen diese raumlichen Beziehungen,

ix

indem sie einen ortsspezifischen Fehlerterm einfuhren und die geographische Abhangigkeit

durch eine Funktion der Distanz zwischen den einzelnen Orten wiedergegeben. Diese Mod-

elle sind allerdings hoch parametrisiert. Hochmoderne Bayes’sche Berechnungen, welche

durch ”Markov chain Monte Carlo” Simulationen implementiert werden, erlauben allerd-

ings deren Modellierung. Eine gebrauchliche Annahme beim geostatistischen Modellieren

ist die der Stationaritat. Das bedeutet, dass die raumliche Korrelation eine Funktion der

Distanz zwischen den Orten ist und nicht der Orte selber. Diese Behauptung gilt allerdings

nicht immer, besonders dann nicht wenn die Malariatransmission uber grosse Entfernun-

gen modelliert werden soll. Deshalb mussen geostatistische Modelle benutzt werden die

zusatzlich die Nicht-Stationaritat berucksichtigen. Das Durchlaufen von geostatistischen

Modellen erfordert mehrfache Inversionen der Varianz-Kovarianz Matrix die die geografis-

che Abhangigkeit darstellt. Fur eine sehr hohe Anzahl von Orten wird dies allerdings als

nicht machbar eingestuft, deshalb werden Methoden zur Optimierung dieser Berechnun-

gen benotigt. Ein weiteres Problem ist, dass die Abhangigkeiten zwischen den okologischen

Faktoren und des Malariarisikos haufig nicht linear sind und daher parametrische Funktio-

nen nicht in der Lage sind die Form dieser Beziehung wiederzugeben. Daher mussen nicht-

parametrische geostatistische Regressionsmodelle, welche den Daten erlauben die Form der

Umwelt-Malaria Beziehung anzunehmen, weiterentwickelt werden um sie fur die Kartierung

von Malaria verfugbar zu machen.

Das Ziel diese Arbeit war es geeignete Modelle fur die Nicht-Stationaritat und grosse

Mengen an geostatistischen Daten zu entwickeln, die sich fur die Malariaepidemiologie

nutzen lassen um exakte Karten der Malariaverteilung zu erstellen. Das Hauptaugen-

merk lag dabei auf der Entwicklung von Methoden fur: (i) die Analyse von nicht sta-

tionaren Malaria Studien; (ii) die Modellierung der nicht linearen Beziehung zwischen

Malariarisiko und Umwelt-/klimatischen Bedingungen; (iii) die Modellierung von geostatis-

tischen Sterblichkeitsdaten welche an sehr vielen Orten gesammelt wurden und (iv) die

Einbeziehung von Unterschieden in der Jahreszeit und dem Alter der Studienteilnehmer

bei der Kartierung von verschiedenartigen Malariastudien.

Kapitel 2 untersucht den raumlichen Effekt von Bettnetzen auf die allgemeine Sterblichkeit

von Kindern durch die Analyse von Daten einer grossen Follow-Up-Studie in einem Ge-

biet mit uber das Jahr konstant hoher Malariaverbreitung in Kilombero Valley, sudlich

x

von Tansania. Die Resultate weisen auf einen fehlenden Gemeinschaftseffekt der Bettnet-

zdichte hin, vermutlich aufgrund der einheitlichen Abdeckung des Studiengebiets mit Net-

zen und der geringen Proportion von erneut behandelten Netzen. Die Sterblichkeitsdaten

dieser Applikation wurden in 7403 Orten gesammelt. Um das Problem der Matrix Inver-

sion zu umgehen wurde ein Bayes’sches geostatistisches Modell entwickelt. Dieses Mod-

ell berechnet den raumlichen Prozess durch eine Untergruppe von Orten und schatzt die

ortsspezifischen Fehlerterme durch die gewichtete Summe der ortsspezifischen Fehlerterme

der Untergruppen durch Wichtungen ab, welche umgekehrt proportional zu der Distanz

der Orte sind.

In Kapitel 3 wurde ein Bayes’sches nicht stationares Model durch die Aufteilung der Studi-

enregion in feste Unterregionen entwickelt. Das Modell beruht auf der Annahme, dass in je-

dem Teilstuck ein fester stationarer raumlicher Prozess ablauft, wobei die Korrelation zwis-

chen den einzelnen Teilstucken mit berucksichtigt wurde. Diese Methode wurde an Daten

zu Malaria aus der MARA Datenbank angewandt und es wurde daraus eine Risikokarte fur

Mali erstellt. Die Vorhersagekraft des nicht stationaren Modells wurde mit der des analo-

gen stationaren Modells verglichen. Die Resultate zeigten dass die Stationaritatsannahme

die Signifikanz okologischer Pradiktoren ebenso wie Abschatzung der raumlichen Faktoren

beeinflusst. Dies deutet im Umkehrschluss an, dass die Annahmen uber den raumlichen

Prozess eine bedeutende Rolle spielen. Modellbewertungen zeigten dabei eine bessere

Vorhersagekraft fur das nicht stationare Modell an. Zusatzlich bestatigten Expertenmei-

nungen, dass die Risikokarte fur Malaria im Falle des nicht stationaren Modells besser

die Lage in Mali widerspiegelt. Diese Arbeit enthullte dass Nicht-Stationaritat eine essen-

tielle Eigenschaft ist, welche unbedingt bei der Erstellung von weiterem Kartenmaterial

fur Malaria berucksichtigt werden sollte.

Kapitel 4 wendet das obere nicht stationare Modell fur die Kartenerstellung zum Malari-

arisiko in Westafrika an, wobei als feste Teilstucke die vier Agrarkulturregionen dienten die

die Region aufteilen. Die Nicht-Linearitat zwischen dem Risiko Malariaparasiten zu haben

und den okologischen Bedingungen wurde beachtet und durch die Benutzung von P-Splines

in der Bayes’schen geostatistischen Modellformulierung vermieden. Das Modell erlaubte

eine separate Malaria-Umwelt Beziehung in jeder Zone. Die Diskontinuitat an den Gren-

zen der Zonen wurde vermieden, indem die raumliche Korrelation durch eine Mischung von

raumlichen Prozessen uber die gesamte Studienflache modelliert wurde. Dabei wurden die

Gewichtungen als exponentielle Funktionen der Distanz zwischen den Ortschaften and den

xi

Zentren der Zonen entsprechend eines jeden raumlichen Prozesses gewahlt.

Der obere Modellansatz ist angebracht fur die Malariakartierung in Gebieten wo eine offen-

sichtliche Teilung (z.B. in okologische Zonen) besteht. Fur Gebiete wo das nicht moglich ist

wurde in Kapitel 5 ein weiteres nicht stationares Modell entwickelt, was den Daten erlaubt

die Anzahl und Gestalt der Teilstucke festzulegen und daher die verschiedenen raumlichen

Prozesse zu bestimmen. Die Unterteilung der Studienflache basierte auf zufalligen Voronoi-

Diagrammen und die Modellparameter wurden mittels ruckwarts verlaufenden Markov

chain Monte Carlo Methoden (RJMCMC) bestimmt beruhend auf der Variablendimension

des Parameterraumes.

In Kapitel 6 wurde die Realisierbarkeit der zuvor entwickelten mathematischen Malaria-

transmissionsmodelle untersucht, welche angepasst wurden an das Alter und die

jahreszeitlichen Schwankungen, indem historische Malariastudien kartiert wurden. Das

Transmissionsmodell wurde angepasst um im Alter der Teilnehmer schwankende Studien-

daten von Mali in eine allgemein gultige Messgrosse fur die Ausbreitungsintensitat zu

uberfuhren. Ein Bayes’sches geostatistisches Modell fur die Abschatzung der Ausbre-

itungsintensitat wurde erstellt, indem als Kovariaten verschiedene umweltbezogene und

klimatische Faktoren genutzt wurden. Bayes’sches Kriging wurde genutzt um gleichmassige

Karten zur Ausbreitung zu erstellen, welche im Weiteren zu altersspezifischen Risikokarten

umgewandelt wurden. Der Modellvergleich mit einigen Testorten zeigte, dass das Trans-

missionsmodell bessere Werte liefert als die direkte Modellierung von Pravalenzdaten.

Dieser Ansatz wurde weiter getestet durch die Analyse der national reprasentativen Daten

der Malaria Indikationsstudie (MIS) in Sambia. Obwohl die MIS Daten nicht die Sel-

ben Einschrankungen haben wie die historischen Daten, lag die Absicht dennoch darin

die Karten zu vergleichenden, die 1) durch die direkte Modellierung der unveranderten

Pravalenzdaten und 2) durch die Modellierung der Daten der Verbreitungsintensitat aus

dem Transmissionsmodell entstanden sind. Beide Karten sagen ahnliche Muster im Malar-

iarisiko voraus, dennoch konnte die Karte basierend auf dem Transmissionsmodell die En-

demizitat ein wenig besser wiedergeben. Die Nutzung von Transmissionsmodellen fur die

Kartierung von Malaria erlaubt die Einbeziehung von jahreszeitlichen Schwankungen und

die Altersabhangigkeit der Malariapravalenz und es beinhaltet alle zur Verfugung stehen-

den historischen Daten die fur die verschiedenen Altergruppen gesammelt wurden.

xii

Acknowledgements

It is my great pleasure to acknowledge many people who contributed to this thesis.

First and foremost, this thesis would not have been possible without the great scientific

support and the remarkable patience of my thesis advisor, Dr. Penelope Vounatsou. Her

enthusiasm and passion for research had motivated and inspired me all these years. I owe

her lots of gratitude for being an outstanding supervisor and a good friend. I could not

have imagined having a better mentor for my PhD.

I also like to express my gratitude to Prof. Thomas Smith for his constructive comments

and for his important support throughout this work. Thank you for being always avail-

able when I needed your advises. Special thanks to Prof. Christian Lengeler for many

productive scientific discussions and to Prof. Don de Savigny for valuable comments and

ideas. I would also like to thank Prof. Mitchell Weiss for a very good and warm working

atmosphere in the department. I am especially grateful to Prof. Marcel Tanner for having

welcomed me at the STI and for creating a stimulating environment for developing my

research.

Many thanks to all the wonderful people in STI, who contributed to a fantastic working

atmosphere. I appreciate the support of Magrit Slaoui, Christine Walliser, Eliane Ghilardi

and Isabelle Bolliger who made my life as a student smooth and pleasant. I wish to thank

the STI library team as well as to the IT support group for their assistance. Special thanks

are addressed to Bianca Pluss for making sure I was getting enough coffee to keep me going,

to Tippi Mak for the good chats, her scientific and friendly advices, to Nadine Riedel for

translating the summary of this thesis in German and for always surprising me with new

learned romanian words, to Claudia Sauerborn for her encouragements and for providing

me with good books, to Amanda Ross for the late night chats in the last phase of our

PhD’s, to Niggi Maire for his help with LaTeX and not only and to Ellen Stamhuis for

xiii

the delicious waffles. A ”grand merci” to Nafomon Sogoba for the lively discussion and

for being a good friend. My warm thanks are addressed to colleagues and friends with

whom I shared the office and good laughter: Daryl Somma, Wilson Sama, Shr-Jie Wang,

Dan Anderegg, Collins Ahorlu, Honorati Masanja, Nadine Koler, Lena Fiebig, Mwifadni

Mrisho, Ricarda Merkle, Josh Yukich, Andri Christen, Gonzalo Duran, Susan Rumisha,

Amina Msengwa. Thanks to those people who provided me with so much whether they

know it or not: Musa Mabaso, Marlies Craig, Michael Bretscher, Melissa Penny, Nakul

Chitnis, Barbara Matthys, Stefan Dongus, Connie Pfeiffer, Manuel Hetzel, Tobias Erlanger,

Giovanna Raso, Elisabetta Peduzzi, Simona Rondini and Monica Daigl.

A BIG thank you to Dora, Lavinia and Romeo who have made Basel a very special place

over all these years. I wish to thank Nicoleta and Marco for the nice week-ends spend

together and for their friendship. My sincerest thanks go to Lucas and a big hug to Anna

and Nadia with whom we shared joyful moments.

The support of many friends back home has been indispensable and I would like particu-

larly to acknowledge Anca, Mihaela, George, Maria, Gabriel, our godparents Mihaela and

Marian Borcan, my sister-in-law Patricia and my brother-in-law Augustin for their con-

stant encouragements throughout this entire journey. I would like to express my warmest

thanks to my parents-in-law for their unconditional support at each turn of the road (”Va

multumesc din suflet”).

I am forever indebted to my parents for the sacrifices made to ensure that I had an ex-

cellent education, for their understanding and endless patience. (”Va voi ramane mereu

indatorata”). My special gratitude is due to my brother for his affection and encourage-

ments when they were most required.

Last but never least, I would like to express my deepest gratitude to Dominic for his endless

love and for the incredible amount of patience he had with me in the last months. Thank

you for accompanying me in this journey.

A lots of other people have contributed in different ways to this thesis. To all of you MANY

THANKS.

This work was supported by the Swiss National Foundation grant Nr.3252B0-102136/1.

xiv

Contents

Summary iv

Zusammenfassung viii

Acknowledgements xii

1 Introduction 1

1.1 Global malaria distribution . . . . . . . . . . . . . . . . . . . . . . . . . . . 2

1.2 Malaria disease and transmission . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.1 The malaria parasite . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.2 The Anopheles vector . . . . . . . . . . . . . . . . . . . . . . . . . . 3

1.2.3 Malaria transmission . . . . . . . . . . . . . . . . . . . . . . . . . . 4

1.2.4 Malaria control interventions . . . . . . . . . . . . . . . . . . . . . . 6

1.3 Mapping malaria transmission . . . . . . . . . . . . . . . . . . . . . . . . . 8

1.3.1 Types of data for malaria mapping . . . . . . . . . . . . . . . . . . 8

1.3.2 Tools for mapping malaria . . . . . . . . . . . . . . . . . . . . . . . 10

1.3.3 Malaria mapping in Africa - a review . . . . . . . . . . . . . . . . . 12

1.3.4 Some methodological issues . . . . . . . . . . . . . . . . . . . . . . 14

1.4 Objectives of the thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 15

2 Spatial effects of mosquito bednets on child mortality 17

2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 19

2.2 Methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.1 Study area and population . . . . . . . . . . . . . . . . . . . . . . . 20

2.2.2 Data collection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 21

2.2.3 Statistical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23

xv

2.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27

2.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 30

3 Bayesian modeling of geostatistical malaria risk data 32

3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 34

3.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.1 Malaria data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 36

3.2.2 Climatic and environmental data . . . . . . . . . . . . . . . . . . . 36

3.3 Bayesian geostatistical models . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.1 Model formulation . . . . . . . . . . . . . . . . . . . . . . . . . . . 38

3.3.2 Bayesian specification and implementation . . . . . . . . . . . . . . 40

3.3.3 Prediction model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.3.4 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 41

3.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 42

3.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 49

3.6 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 52

4 Mapping malaria risk in West Africa using a Bayesian nonparametric

non-stationary model 53

4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

4.2 Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

4.3 Spatial modeling . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

4.3.1 Non-linearity of covariates effect . . . . . . . . . . . . . . . . . . . . 60

4.3.2 Spatial correlation and non-stationarity . . . . . . . . . . . . . . . . 61

4.3.3 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4.3.4 Model validation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63

4.3.5 Implementation details . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 65

4.5 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 74

5 Non-stationary partition modeling of geostatistical data for malaria risk

mapping 78

5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

5.2 Motivating example: mapping malaria risk in Mali . . . . . . . . . . . . . . 82

5.3 Modeling non-stationarity via dependent spatial processes . . . . . . . . . 83

xvi

5.4 Implementation details . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.4.1 Model fit . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84

5.4.2 Prediction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.5 Analysis of the malaria prevalence data . . . . . . . . . . . . . . . . . . . . 87

5.6 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 92

6 Mapping malaria using mathematical transmission models 94

6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 96

6.2 Materials and methods . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.2.1 Malaria data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.2.2 Environmental data . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

6.2.3 Statistical analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

6.3 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 103

6.4 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 114

6.5 Appendix . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

7 General discussion and conclusions 119

Bibliography 127

xvii

xviii

List of Figures

1.1 Geographic distribution of malaria . . . . . . . . . . . . . . . . . . . . . . 2

2.1 Distribution of the DSS households according to their socio-economic status 21

3.1 Sampling locations of the MARA surveys in Mali. . . . . . . . . . . . . . . 37

3.2 The distribution of Bayesian p-values for the stationary model and the non-

stationary ones with fixed number of tiles in Mali . . . . . . . . . . . . . . 43

3.3 The distribution and the sum Tχ2 of the χ2-values over the test points in Mali 44

3.4 Map of predicted malaria risk for south Mali using the stationary model. . 47

3.5 Map of predicted malaria risk for south Mali using the non-stationary model

with 2 fixed tiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 47

3.6 Map of prediction error for south Mali using the stationary model. . . . . . 48

3.7 Map of prediction error for south Mali using the non-stationary model with

2 fixed tiles. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

4.1 Sampling locations of the MARA surveys West Africa . . . . . . . . . . . . 58

4.2 The distribution of Bayesian p-values and Kulback-Leibler difference mea-

sure for the two non-stationary Bayesian geostatistical models in West Africa 65

4.3 Percentage of observed locations with malaria prevalence falling in the cred-

ible intervals of the posterior predictive distribution in West Africa . . . . 66

4.4 Estimated non-linear effect (P-spline) of environmental factors on malaria

risk in West Africa, Sahel . . . . . . . . . . . . . . . . . . . . . . . . . . . 68


risk in West Africa, Sudan Savanna . . . . . . . . . . . . . . . . . . . . . . 69


risk in West Africa, Guinea Savanna . . . . . . . . . . . . . . . . . . . . . 70

xix


risk in West Africa, Equatorial Forest . . . . . . . . . . . . . . . . . . . . . 71

4.8 Map of predicted malaria prevalence in children 1− 10 years for West Africa 74

4.9 Map of prediction error for West Africa . . . . . . . . . . . . . . . . . . . . 75

5.1 Sampling locations in sub-Saharan Mali . . . . . . . . . . . . . . . . . . . . 82

5.2 Posterior distribution of the number of tiles in Mali . . . . . . . . . . . . . 89

5.3 Percentiles (2.5th, 50th and 97.5th) of the posterior distribution for the

spatial parameters in Mali . . . . . . . . . . . . . . . . . . . . . . . . . . . 90

5.4 Predicted malaria prevalence in sub-Sahara Mali . . . . . . . . . . . . . . . 91

5.5 Map of prediction error for sub-Sahara Mali . . . . . . . . . . . . . . . . . 91

6.1 The distribution of Bayesian p-values for the geostatistical model based on

the raw prevalence data and for the geostatistical intensity model . . . . . 103


ible intervals of the posterior predictive distribution in Mali. . . . . . . . . 104

6.3 Estimated effect (P-spline) of environmental factors on EIR in Mali . . . . 105

6.4 Estimated effect (P-spline) of environmental factors on EIR in Zambia . . 106

6.5 Predicted annual entomological inoculation rate (EIR) in Mali . . . . . . . 108

6.6 Prediction error of annual entomological inoculation rate (EIR) in Mali . . 108

6.7 Predicted malaria prevalence for children under 5 years old in Mali . . . . 109

6.8 Predicted malaria prevalence for children between 1 and 10 years old in Mali 109

6.9 Predicted annual entomological inoculation rate (EIR) in Zambia . . . . . 110

6.10 Prediction error of annual entomological inoculation rate (EIR) in Zambia 111

6.11 Predicted malaria prevalence for children under 5 years old in Zambia . . . 112

6.12 Predicted malaria prevalence for children under 5 years old in Zambia esti-

mated by directly analyzing the prevalence data . . . . . . . . . . . . . . . 113

xx

List of Tables

2.1 Overall and district-specific child mortality rates in Kilombero Valley, Tan-

zania . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.2 Estimates of the effect of bednet measures on in Kilombero Valley, Tanzania 25

2.3 Association of child mortality with sex, socio-economic status, bednet den-

sity at household level and distance to nearest health facility in Kilombero

Valley, Tanzania . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.4 Estimated effect of bednet measures on mortality of children without nets

in Kilombero Valley, Tanzania . . . . . . . . . . . . . . . . . . . . . . . . . 27

3.1 Spatial databases used in the spatial analysis in Mali . . . . . . . . . . . . 36


ible intervals of the posterior predictive distribution in Mali . . . . . . . . . 44

3.3 Parameter estimates for geostatistical models in Mali . . . . . . . . . . . . 46

4.1 Measures of environmental predictor used in the analysis of malaria risk

data in West Africa . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 64

4.2 Posterior estimates for land use coefficients in West Africa . . . . . . . . . 72

4.3 Posterior estimates of spatial parameters in West Africa . . . . . . . . . . . 73

5.1 Posterior estimates for environmental covariate effects in Mali . . . . . . . 89

6.1 Age groups of the population included in the MARA surveys in Mali between

years 1962-2001. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

6.2 Spatial databases used in the spatial analysis in Mali. . . . . . . . . . . . . 100

6.3 Spatial databases used in the spatial analysis in Zambia. . . . . . . . . . . 100

6.4 Posterior estimates of spatial parameters in Mali and Zambia . . . . . . . . 107

xxi

xxii

Chapter 1

Introduction

2 Chapter 1. Introduction

1.1 Global malaria distribution

Malaria is the most important tropical disease, remaining widespread throughout the trop-

ical and subtropical regions, including parts of Africa, Asia and Americas (Figure 1.1). It is

a major cause of illness and death in large areas of the developing world, especially Africa.

According to the World Malaria Report (WHO, 2005), at the end of 2004 there were 107

malaria endemic countries and 3.2 billion people were at risk of malaria. Malaria causes

at least 300 million and possibly as many as 500 million cases of acute illness each year,

which result in 1-3 million deaths (Breman et al., 2004). Ninety percent of deaths occur

in sub-Sahara Africa. The large majority of these deaths are in children younger than

five years of age, being estimated that every 30 seconds a child dies because of malaria.

These rough estimates are not reliable because of the inadequate malaria case reporting in

most endemic countries and lack of national wide malaria surveys. Accurate estimates of

the burden of disease are required for planning, implementation and evaluation of malaria

control programs. Hence, there is an urgent need for precise estimates of the number of

people at risk of malaria to optimize the use of limited resources in high-risk areas.

Figure 1.1: Geographic distribution of malaria.

1.2 Malaria disease and transmission 3

1.2 Malaria disease and transmission

1.2.1 The malaria parasite

Malaria is an infectious disease caused by four parasitic protozoa of the genus Plasmodium,

namely: P. falciparum, P. vivax, P. ovale and P. malariae. The human malaria parasites

require the presence of two hosts to complete their life cycle: the human, which represents

the intermediate host and the female mosquito, which is the defitinive host. In mosquito,

the parasite reproduces sexually by combining sex cells, whereas in human the parasite

reproduces asexually, first in liver cells and then in red blood cells.

A human infection begins when an infected female mosquito takes a blood meal, passing

sporozoites into the human’s bloodstream. The sporozoites travel with the blood to the

liver and enter the liver cells. There, the sporozoites mature into schizonts, which rupture

and release merozoites into the blood stream. Merozoites invade the red blood cells, where

they undergo the second asexual reproduction into human, forming again schizonts. When

the schizonts mature, the cell rupture and merozoites burtst out, infecting other red blood

cells. This repeating cycle depletes the body of oxygen and causes fever, triggering the

onset of disease symptoms. In the red blood cells some merozoites develop into sex cells

known as male and female gametocyte. When a female mosquito bites an infected human

the gametocyte are ingested. In mosquito’s stomach the gametocyte undergo sexual repro-

duction, forming a zygote. The zygote multiplies to form sporozoites, which make their

way to the mosquito’s salivary glands. Inoculation of the sporozoites into a new human

host perpetuates the malaria life cycle.

1.2.2 The Anopheles vector

The distribution of the malaria parasite is largely determined by the distribution of the

mosquito vectors which, in the case of malaria parasites of humans, are all of the genus

Anopheles. There are 430 Anopheles species, of which around 70 are malaria vectors, but

only 40 of these are thought to be of major public health importance (Service and Townson,

2002). Among these, the An. gambiae complex and An. funesus are the primary malaria

vectors in Africa. An. gambiae s.s. and An. arabiensis are the most widely distributed

species of the An. gambiae complex in sub-Saharan Africa. Although these sibling species

are morphologically indistinguishable, they exhibit different behavioral attributes. An.

gambiae s.s. is predominant in humid areas, prefers feeding on humans (anthropophilic)


and rests mainly indoors. On the other hand, An. arabiensis is more tolerant in the

drier savanna regions, it often feeds on animals (zoophilic) and rests outdoors. Both

species breed in temporary habitats such as pools, puddles, rice fields. An. funesus prefers

permanent water bodies with vegetation such as swamps and marshes, feeds both indoors

and outdoors, mainly on humans and rests indoors.

1.2.3 Malaria transmission

Measures of malaria endemicity and transmission

Measures of malaria transmission quantify malaria risk and endemicity levels and they are

the basis of decision making in malaria control. These measures include parameters related

to malaria transmission from mosquito to humans (i.e. entomological inoculation rates,

force of infection, incidence rates, parasite prevalence) and parameters related to malaria

vectors (i.e. mosquito survival, infection probability).

The most commonly used measure of malaria endemicity is the prevalence of human

infections within a community. Information on malaria prevalence is collected through

community-based surveys by computing the percentage of individuals found with a posi-

tive blood slide. In 1950s WHO classified malaria endemicity using both the parasite and

the spleen rates (percent of children with enlarged spleen) as: hypoendemic, mesoendemic,

hyperendemic and holoendemic. Malaria prevalence of the same population may vary in

time, depending on the seasonality and stability of the disease.

Entomological inoculation rate (EIR) is the most used measure for assessing malaria trans-

mission intensity. It represents the number of infective mosquito bites an individual is

likely to be exposed to over a defined period of time, usually 1 year. EIR is expressed as

the product of the anopheline mosquito density, the average number of mosquitoes biting

each person in one day and the proportion of infective mosquitoes (sporozoite rate). The

product of the first two measures is known as human biting rate and is assessed using

techniques like human bait catch, pyrethrum spray collection and light trap catch. The

sporozoite rate is determined by dissection and examination of mosquito salivary glands or

by the enzyme-linked immunosorbent assay (ELISA), a technique with high sensitivity and

species specificity. Measurements of EIRs during longitudinal studies provides information

on seasonal variations in transmission. A review of EIR estimates across Africa (Hay et


al, 2000) found the mean annual EIR of 121 infective bites, ranging from a minimum of 0

in Burkina Faso, The Gambia and Senegal to a maximum of 884 in Sierra Leone.

Incidence of malaria is a direct measure of the amount of malaria transmission because

it represents the number of new malaria cases diagnosed during a given time interval

in relation to the unit of population in which they occur. Incidence data are usually

collected in health facilities. In most settings in sub-Sahara Africa is not possible to

perform laboratory confirmation of malaria diagnoses, therefore incidence of fever is used

as a proxy for incidence of malaria.

Force of infection is the rate at which susceptible individuals become infected by malaria

parasite. These data are an alternative to malaria incidence data which are difficult to be

collected in communities where the prevalence of infection reached saturation. MacDonald

was the first to propose an estimate of force of infection using infant parasite conversion

rates for malaria.

The basic reproductive rate R0 quantifies the transmission potential and is defined as the

average number of successful offspring that a parasite is intrinsically capable of producing.

Other two important parameters related to malaria vectors are the mosquito survival per

gonotrophic cycle,P and the infection probability (sometimes called infectious reservoir)

K, that is the probability that a mosquito becomes infected when it takes a feed. These

quantities cannot be determined directly from field data, therefore transmission models are

needed to estimate them.

Determinants of malaria transmission

Malaria transmission is affected by different factors like environmental conditions, poverty

(socio-economic status), population movement (migration, urbanization), limited access to

health services, poor quality of the public health services or water management methods

(e.g. irrigation, dam constructions) that increase the mosquitoes population near human

habitats. Climate is the main driver of malaria transmission with climate variability influ-

encing the level of transmission intensity.

The amount and duration of malaria transmission is influenced by the ability of parasite and

mosquito vector to co-exist long enough to enable transmission to occur. The distribution

and abundance of the parasite and mosquitoes population are sensitive to environmental

factors like temperature, rainfall, humidity, presence of water and vegetation.


Temperature plays an important role in the distribution of malaria transmission by influ-

encing both the parasite and the vector. In particular, it has an effect on the survival of the

parasite in the Anopheles mosquito. Optimum conditions for the extrinsic development of

malaria parasite are between 25◦C and 30◦C, but as the temperature decreases, the number

of days necessary to complete the extrinsic phase increases. At temperatures below 16◦C

the sporogonic cycle stops. For the vector, temperature affects the development rate of

mosquito larvae and the survival rate of adult mosquitoes. Mosquitoes generally develop

faster and feed earlier in their life cycle and at a higher frequency in warmer conditions.

Development from egg to adult may occur in 7 days at 31◦C, but takes about 20 days at

20◦C.

Rainfall is one of the major factors influencing malaria transmission. It provides breeding

sites for mosquitoes to lay their eggs, increasing the vector population and it increases

humidity, improving mosquitoes survival rate. When humidity is below 60% the longevity

of mosquitoes is drastically reduced. Mosquitoes are usually found in areas with annual

average rainfall between 1100 mm and 7400 mm. However, excessive rain can have the

opposite effect, by impeding the development of mosquito eggs or larvae, by flushing out

many larvae and pupae out of the pools or by decreasing the temperatures, which can stop

malaria transmission in areas at high altitudes.

Vegetation type and the amount of green vegetation are important factors in determining

mosquito abundance by providing feeding provisions and protection from climatic condi-

tion but also by affecting the presence or absence of the human hosts and therefore the

availability of blood meals.

Land use changes may influence climatic conditions like temperature or evapotranspira-

tion (Patz et al., 2005), which are main determinants of the abundance and longevity of

mosquitoes. In addition, agriculture practices and human-made environmental alterations

could influence the malaria vector population.

1.2.4 Malaria control interventions

In recent years several initiatives have been launched to tackle malaria in various parts of

the world. In 1998 WHO initiated the Roll Back Malaria (RBM) Partnership with the

goal of reducing malaria burden by at least 50% by the year 2010, applying evidence-based


intervations through strengthened health services. The Global Fund to Fight AIDS, Tuber-

culosis and Malaria (GFATM) was established in 2002, giving malaria-endemic countries

access to additional external funding for malaria control. In 2005 President Bush launched

the President’s Malaria Initiative (PMI), a five-year programme which aims to reduce

deaths due to malaria by 50% in 15 African countries.

Control measures are directed at each component involved in the malaria transmission

cycle: the human host, the parasite and the mosquito vector. Complete cure of clinical

malaria requires treatment with several drugs over several days and this creates problems

of costs and compliance. Prophylaxis drugs have been of great benefit and widely used

as a measure of malaria control, but they are no longer effective in many tropical areas

because the parasite developed resistance to drugs.

Vector control remains, in general, the most effective tool to prevent and control malaria

transmission. The principal objective of vector control is to reduce malaria morbidity and

mortality by reducing the levels of transmission. Common measures include indoor and

outdoor house insecticide spraying, the use of insecticide treated nets (ITN) and environ-

mental measures such as management of water bodies and vegetation clearance. Applica-

tions of these techniques, alone or in combination, reduce human-mosquito contact, vector

abundance and vector infectivity. ITN’s are increasingly being promoted as a an efficient

method for reducing the burden of malaria. In trials conducted between 1980 and 2000

ITN’s were shown to reduce childhood malarial deaths in endemic areas in Africa by 17%

and roughly halve the number of clinical malaria episodes (Lengeler, 2004). Although the

use of ITN’s provide significant individual effect, it still remains unclear what are the effects

of ITN’s on the wider community of bednet users and non-users.

Malaria control is a dynamic process which depends on the local epidemiological situation

and of the facilities and resources available. Therefore, it is important that maps of malaria

transmission are available for guiding control measures to high risk areas. These maps can

also provide a baseline to evaluate the effectiveness of interventions programs.


1.3 Mapping malaria transmission

Although a lot of effort and resources have been put into the control of malaria, reliable

estimates and mapping of malaria burden are not available. Maps of malaria distribution

are valuable in increasing the effectiveness of decision-making. Maps are also useful to

assess the effect of intervention programs by estimating the malaria distribution prior

and post interventions. Recently, there is a renewed interest in mapping malaria (Snow

et al., 2005; Hay et al., 2006) as well as different efforts in assembling existing malaria

data (WHO, 2007; Malaria Atlas Project (MAP) (Guerra et al., 2007); MARA project

newly funded by Bill and Melinda Gates foundation in 2007; Mekong malaria (Socheat

et al, 2003)). Malaria is an environmental disease since its transmission depends upon

the distribution and abundance of the mosquitoes, which are sensitive to climate. Hence,

mapping malaria distribution is based on availability of malaria and environmental data

as well as appropriate methods to analyze these data.

1.3.1 Types of data for malaria mapping

The main sources of data for mapping malaria are the following.

(i) historical malaria prevalence data. The ”Mapping Malaria Risk in Africa” (MARA/ARMA,

1998) represents the most comprehensive database on malaria data in Africa. It contains

malaria prevalence data collected over 10, 000 geographically positioned surveys from gray

or published literature across the whole continent. The project was initiated over a decade

ago to provide comprehensive, empirical and standardized maps of malaria distribution

and endemicity in Africa. The database is currently being updated. A parallel project for

assembling historical malaria data is being carried out by the MAP (Guerra et al., 2007).

Maps from historical data may not reflect the current malaria situation at a given location,

which could be influenced by control measures. Unfortunately, information on historical

interventions is not generally available, therefore it is not possible to account for it. The

limited number of surveys during the recent years requires inclusion of surveys from earlier

years for mapping purposes. On the other hand, historical data are useful for looking at

temporal changes of the malaria situation. The major drawback of these type of data is the

heterogeneity in season and age since they are collected at non-standardized seasons and

include overlapping age groups of the population. In addition, the data are sparse in time

and space. These constraints make it difficult to consider seasonality and age adjustment

1.3 Mapping malaria transmission 9

in malaria mapping (Gemperli et al., 2005).

(ii) parasite prevalence data derived from nationally representative surveys. These type of

data are useful in countries with high transmission intensity and are collected form Demo-

graphic and Health surveys (DHS), Multiple Indicator Cluster Surveys (MICS) and Malaria

Indicator Surveys (MIS). Zambia successfully conducted the first nationally representative

household survey assessing coverage of malaria interventions and malaria-related burden

among children under five years of age. The survey was conducted during May and June

2006 and was led by the Ministry of Health through the National Malaria Control Center

(NMCC) in collaboration with many Zambia RBM partners. The MIS included informa-

tion on intervention (IRS and ITNs) coverage, morbidity and background characteristics

(i.e. household assets). The goal of the project was to provide baseline data against which

to measure progress toward achieving its goals set forth in the National Malaria Strategic

Plan for 2006-2010.

(iii) clinical malaria incidence data. These data are appropriate in areas with low malaria

risk, like many countries in South-East Asia, since it is unlikely for people in the community

to tolerate the parasite without being sick. Incidence data depend on precise estimates of

the population at risk of malaria. Many countries in Africa do not have a reliable disease

surveillance system, therefore routine malaria statistics can not be used (exception: South

Africa, Zimbabwe, Botswana, Namibia). The major drawback of this type of data is that

for countries with limited access to laboratory confirmation of cases - like most of the

countries in sub-Sahara Africa - the new malaria cases refer to patients who are suspected

to have malaria based on clinical signs and symptoms.

(iv) entomological data. They provide direct measures of malaria transmission via esti-

mates of EIR, sporozoite rates and other vector-related parameters. However, the data

collection methods are not standardized, therefore the estimated transmission parameters

could differ widely, depending on the techniques used. In regions with low malaria trans-

mission the number of mosquitoes (infected mosquitoes) is very low, so the sampling error

will be large. Because continuous collections of mosquitoes over a long period of time is

difficult, the entomological data are usually derived from short/medium-term studies over

small areas, hence prediction of these data at unsampled locations is difficult (Hay et al.,

2000).

Recently, incidence and entomological data were both combined by Mabaso (2007) for

mapping seasonality of malaria transmission in Africa. Seasonal dynamics of malaria


transmission are important for timing malaria control and preventive strategies, as well

as for mapping malaria transmission via malaria transmission model (Gemperli et al.,

2005; Gemperli et al., 2006).

1.3.2 Tools for mapping malaria

GIS and remote sensing

Remote sensing (RS) imagery is a powerful tool for determining the environmental predic-

tors of malaria transmission. It is an important source that can provide such spatially rich

information.

In recent years, significant progress has been made in the development of geographic infor-

mation systems (GIS) and their applications in public health and spatial epidemiology. GIS

are computerized systems capable of collecting, storing, handling, analyzing and display-

ing all forms of geographically referenced information. In GIS, information from different

sources are represented as layers and linked in a spatial context. However, further research

is needed on the relation between satellite-derived proxies on environmental conditions and

ground climate data.

Since 1990s RS and GIS provide useful tools for mapping malariological indicators in Africa.

Craig et al. (1999) produced a climatic suitability map of malaria transmission in sub-

Sahara Africa and Snow et al. (1999) estimated the number of people at risk of malaria

worldwide, by continent. In addition, integrated RS and GIS were used to produce maps of

malaria vector distribution (Coetzee et al., 2000) and maps of vector breeding sites (Wood

et al., 1992).

Malaria mapping is based on estimating the relation between malaria transmission and

environmental/climatic factors and using this relation to predict malaria transmission at

locations where the information is not available. Although the integrated GIS and remote

sensing are valuable tools, they are not able to quantify the malaria-climate relation and to

produce model-based predictions. Some GIS softwares have limited statistical capabilities,

but these are inadequate for analyzing prevalence survey data.


Statistical modeling

Statistical models are useful for quantifying the relation between malaria risk and environ-

mental factors and upon this relation predicting malaria risk at locations without observed

malaria data. Malariological data are correlated in space since locations in close proximity

have similar risk. Standard statistical methods assume independence of the observations

and are not appropriate for analyzing spatially correlated data because they underestimate

the standard error and thus the significance of the risk factors would be overestimated (Ver

Hoef et al. 2001).

The type of spatial statistical methods used to analyze spatially correlated data depends

on the nature of the geographical information. There are three kinds of spatial data:

point-level (geostatistical), areal (lattice) and point patterns. Geostatistical data arise

from observations collected at fixed locations over a continuous study region. Analysis of

geostatistical data aims to identify environmental factors that determine the distribution

of malaria in the presence of spatial correlation and kriging, that is spatial prediction at

unobserved locations. Geographical dependence is considered as a function of the distance

between locations. Areal data usually consist of counts or rates aggregated over a par-

ticular set of contiguous units. The focus of the analysis is to identify spatial patterns

or trends and to assess association between malaria data and environmental factors that

vary gradually over geographical regions. Spatial proximity is defined by a neighboring

structure. Point pattern data arise when the locations of particular events are not fixed,

but random quantities. Questions of interest with these data center on whether events

appear sporadically or they are clustered and which are the risk factors associated with

such clusters.

Exploratory tools (variogram for geostatistical data, Moran’s I and Geary’s C for areal

data and clustering statistics for point pattern data) describe the geographical pattern of

the data and are available in most statistical packages. However, they are unable to filter

the noise present in the data due to variable sample size between locations and produce

smooth maps highlighting disease patterns.

Spatial models introduce at each data location (in the case of geostatistical data) or at each

area (in the case of lattice data) an additional parameter on which the spatial correlation

is incorporated. Hence these models are highly parametrized and fitting them could be

challenging. Methods based on the maximum likelihood approaches are not appropriate


because they are not able to estimate simultaneously the malaria-climate relation and the

spatial correlation. Bayesian hierarchical approaches avoid the computational problems

in likelihood-based fitting by relying inference on Markov chain Monte Carlo (MCMC)

simulation methods, hence are the best alternative in analyzing spatially correlated data.

For areal data, the mostly used prior distribution for random effects are simultaneously

autoregressive (SAR) models (Whittle, 1954), conditional autoregressive (CAR) models

(Clayton and Kaldor, 1987) and multivariate CAR models for multinomial response data

(Vounatsou et al., 2000). In malaria epidemiology these approaches were employed to

map malaria vector densities in a single village in Tanzania (Smith et al., 1995), malaria

incidence rates in KwaZulu-Natal, South Africa (Kleinschmidt et al., 2001b) and in the

state of Para, Brazil (Nobre et al., 2005) and to study malaria seasonality in Zimbabwe

(Mabaso et al., 2005).

In the case of geostatistical data, the random effects model the underlying spatial process

via a multivariate Normal distribution with the covariance matrix defined as a function

of the distance between locations. Geostatistical models were introduced by Diggle et al.

(1998) and have been employed to map malaria transmission in The Gambia (Diggle et

al., 2002), Mali (Gemperli et al., 2005) and West-Africa (Gemperli et al., 2006).

Bayesian hierarchical models have become powerful methods in modeling spatial data due

to development of simulation techniques like MCMC (Gelfand and Smith, 1990). These

methods are employed to derive empirical approximation of the posterior distribution of

parameters. Well-known methods include: Metropolis-Hastings algorithm (Metropolis et

al., 1953; Hastings, 1970), the Gibbs sampler algorithm (Gelfand and Smith, 1990) and

reversible Jump MCMC (Green, 1995).

1.3.3 Malaria mapping in Africa - a review

The first global map of malaria endemicity was produced by Lysenko and Semashko al-

most 40 years ago (Lysenko and Semashko, 1968). The map combines data derived form

historical documents and maps of several malariometric indices with expert opinion and

simple climatic/geographical iso-lines. Historical maps at country level include the map of

Namibia (de Meillon, 1951), Tanzania (Wilson, 1956), Kenya (Nelson, 1959) and Botswana

(Chayabejata et al., 1975). The major drawback of the historical maps is that they made


limited use of empirical evidence and they did not capture the spatial and temporal het-

erogeneity of malaria transmission.

Hay et al. (1998) analyzed RS-derived climatic and malaria admission data recorded over

5 years in Kenya to produce a map of malaria seasonality. Thomson et al. (1999) used RS

surrogate of climate to predict the number of children infected with P.falciparum according

to different levels of bednet usage in The Gambia. Craig et al. (1999) used GIS techniques

to spatially interpolated weather station data and based on the assumption that malaria

transmission at continental level is limited by temperature and rainfall, defined climatic

suitability for malaria transmission in sub-Sahara Africa. Using the malaria distribution

model and a population distribution model (Deichmann, 1996), Snow et al. (1999) esti-

mated the number of people at risk of malaria in sub-Sahara Africa. A spatial statistics

approach was used by Kleinschimdt et al. (2000) and Kleinschimdt et al. (2001) to map

malaria risk in Mali and West Africa, respectively, by fitting a standard regression model

and applying classical kriging on the model residuals. Rogers et al. (2002) investigated

the factors that influence the dynamics of malaria vector population and transmission and

produce a map of EIR in Africa. Tanser el al. (2003) predicted potential effect of climate

change on malaria transmission in Africa. Omumbo et al. (2005) defined two ecologi-

cal zones in East Africa and modeled malaria risk using high-spatial resolution satellite

imagery, as well as urbanization, water bodies and land use parameters.

The pioneers in using Bayesian statistics in spatial epidemiology of malaria were Diggle

et al. (2002) who fitted a geostatistical model on malaria survey data from The Gambia

without having produced a malaria risk map. Gemperli et al. (2005) and Gemperli et al.

(2006) produced maps of malaria transmission in Mali and West Africa, respectively, mak-

ing use of the Garki transmission model and Bayesian kriging. Gemperli (2003) was also

the first to consider the non-stationarity feature of malaria and map malaria risk in Mali.

Sogoba et al. (2007) fitted Bayesian geostatistical models to identify the environmental

determinants of the relative frequencies of An. gambiae s.s. and An. arabiensis mosqui-

toes species and to produce smooth maps of their spatial distribution in Mali. Using the

Zambia National MIS data, Riedel et al. (unpublished) mapped malaria risk in Zambia.


1.3.4 Some methodological issues

There are a number of methodological problems related with malaria modeling, that is: i)

modeling very large non-Gaussian geostatistical data; ii) analysis of non-stationary non-

Gaussian geostatistical data; iii) modeling the non-linear effect of environmental/climatic

factors on malaria risk and iv) adjusting for age and seasonality.

Fitting geostatistical models for non-Gaussian data requires repeated inversions of the

covariance matrix of the spatial random effects which, for very large number of locations

(N), is not feasible. In spatial data analyzes this computational challenge is referred to

as the ”large N problem”. A number of strategies have been suggested for handling large

spatial data sets (Gelfand et al., 2000; Rue and Tjelmeland, 2002; Stein et al., 2004;

Gemperli and Vounatsou, 2006; Paciorek, 2007). Xia and Gelfand (2005) proposed an

approach based on the assumption that a spatial random process can be approximated by

a linear combination of M << N random variables. This approach has the advantage of

avoiding the inversion of the large NxN covariance matrix by reducing the problem to the

inversion of a much smaller size matrix MxM .

Most applications of geostatistical models assume that the spatial correlation is a function

of the distance and independent of locations, that is the spatial process is stationary. This

hypothesis is not appropriate when malaria data are analyzed since local characteristics

influence the spatial structure differently at various locations. There are a number of

approaches for modeling non-stationarity in statistical literature (Haas, 1995; Higdon et

al., 1998; Fuentes and Smith, 2002; Sampson and Guttorp, 1992). However, the most

attractive method is the one developed by Kim et al. (2005) who modeled non-stationarity

by partitioning the study area in random tiles, assuming an independent stationary process

in each tile and independence between tiles. The only reference to non-stationarity in

malaria mapping is by Gemperli (2003) who extended the work of Kim et al. (2005) for non-

Gaussian data. The tessellation approach tackles another issue in Bayesian geostatistical

modeling, that is computation of the inverse of the covariance matrix of the spatial process

which appears in the prior distribution of the random effects. When the number of data

locations is very large matrix inversion may not be feasible within time constraints.

The relation between malaria transmission and climate is complex and often non-linear.

Nonparametric regression methods relax the assumption of linearity and the relation be-

tween outcome variable and the associated predictor variables is determined by the data,

1.4 Objectives of the thesis 15

not by a pre-specified model like in the parametric case. There are a lot of nonparame-

tric modeling alternatives; here we mention local polynomial regression (Cleveland, 1979),

kernel smoothing (Silverman, 1986), fractional polynomials (Royston and Altman, 1994)

and spline smoothing. Splines are flexible models that take the form of piecewise poly-

nomials joined at knots, where continuity constraints are imposed so that the function is

smooth. Spline smoothing methods include regression splines (Eubank, 1988), B-splines

(de Boor, 1978) and penalized splines (Eiler and Marx, 1996). The latter was implemented

in the Bayesian framework by Crainiceanu et al. (2005), allowing simultaneous estimation

of smooth functions and smoothing parameters. In malaria epidemiology field the most

popular methods for modeling non-linearity are the use of the predictors in categories or

functional transformations of the predictors (Kleinschmidt et al, 2001a; Gemperli et al.,

2006).

Malaria is seasonal and age dependent, therefore it is important when modeling survey data

to account for seasonality and adjust for age. This task becomes challenging when analyzing

historical field survey data because they were collected in different seasons and at non-

standardized and overlapping age groups of the population. Gemperli et al. (2005; 2006)

demonstrated the feasibility of using malaria transmission models in malaria risk mapping

by employing the Garki malaria transmission model (Dietz et al, 1974) to convert observed

prevalence data into an estimated entomological measure of transmission intensity. They

fitted a Bayesian geostatistical model on the estimates of EIR and employed Bayesian

kriging to obtain a smooth map of EIR. The transmission model was applied again to

convert the predicted EIR values into estimates of malaria prevalence for specific age

groups of the population.

1.4 Objectives of the thesis

The overall objectives of this thesis were: 1) to develop Bayesian models for the analysis of

Gaussian and non-Gaussian (binomial and negative binomial) geostatistical non-stationary

data and 2) to validate and implement this methodology in the field of malaria epidemiol-

ogy to produce maps of malaria risk and malaria transmission intensity and to assess the

spatial effect of malaria control interventions on child mortality.


The specific methodological objectives were:

(i) development of geostatistical models for negative binomial data which allow applications

to large data sets (Chapter 2);

(ii) development and validation of methods for non-stationary prevalence data appropriate

for malaria mapping (Chapter 3 and Chapter 5);

(iii) modeling non-linear relation between malaria risk and environmental predictors (Chap-

ter 4);

(iv) development and validation of models for mapping malaria transmission intensity

(Chapter 6).

The above mentioned models were applied on data extracted from the Demographic Surveil-

lance System (DSS), MARA and Malaria Indicator Surveys (MIS) databases to:

(1) evaluate the spatial effect of bednet use on child mortality in Kilombero Valley, Tan-

zania;

(2) identify environmental predictors of malaria transmission and produce smooth maps of

malaria risk in Mali;

(3) produce smooth maps of malaria risk in West Africa based on the non-linear relation

between climate and malaria risk;

(4) produce age and seasonality adjusted malaria risk maps from heterogeneous malaria

survey data in Mali and Zambia.

Chapter 2

Spatial effects of mosquito bednets

on child mortality

Gosoniu L.1 ,Vounatsou P.1, Tami A.1,2, Nathan R.3, Grundmann H.4, Lengeler C.1

1 Swiss Tropical Institute, Basel, Switzerland2 Royal Tropical Institute, Biomedical Research, Amsterdam, The Netherlands3 Ifakara Health Research and Development Center, Ifakara, Tanzania4 Centre for Infectious Diseases Epidemiology, National Institute for Public Health and the

Environment, Bilthoven, The Netherlands

This paper has been published in BMC Public Health 2008, 8:356

18 Chapter 2. Spatial effects of mosquito bednets on child mortality

Summary

Background: Insecticide treated nets (ITN) have been proven to be an effective tool

in reducing the burden of malaria. Few randomized clinical trials examined the spatial

effect of ITNs on child mortality at a high coverage level, hence it is essential to better

understand these effects in real-life situation with varying levels of coverage. We analyzed

for the first time data from a large follow-up study in an area of high perennial malaria

transmission in southern Tanzania to describe the spatial effects of bednets on all-cause

child mortality.

Methods: The study was carried out between October 2001 and September 2003 in 25

villages in Kilombero Valley, southern Tanzania. Bayesian geostatistical models were fitted

to assess the effect of different bednet density measures on child mortality adjusting for

possible confounders.

Results: In the multivariate model addressing potential confounding, the only measure

significantly associated with child mortality was the bed net density at household level;

we failed to observe additional community effect benefit from bed net coverage in the

community.

Conclusions: In this multiyear, 25 village assessment, despite substantial known inade-

quate insecticide-treatment for bed nets, the density of household bed net ownership was

significantly associated with all cause child mortality reduction. The absence of community

effect of bednets in our study area might be explained by (1) the small proportion of nets

which are treated with insecticide, and (2) the relative homogeneity of coverage with nets

in the area. To reduce malaria transmission for both users and non-users it is important

to increase the ITNs and long-lasting nets coverage to at least the present untreated nets

coverage.

2.1 Introduction 19

2.1 Introduction

Plasmodium falciparum malaria is a leading infectious disease, accounting for approxi-

mately 300 to 500 million clinical cases each year and causing over one million deaths,

mostly in African children younger than 5 years. Insecticide treated nets (ITN) have been

proven to be an effective tool in reducing the burden of malaria (D’Alessandro et al., 1995;

Binka et al., 1996; Nevill et al., 1996). Numerous trials all over the world have shown

that such nets can reduce child mortality in endemic areas in Africa by 17% and roughly

halve the number of clinical malaria episodes (Lengeler, 2004). These results were later

confirmed under programme implementation (D’Alessandro et al., 1995; Schellenberg et

al., 2001). It is well known that the use of ITNs provides significant individual protection,

but direct and indirect effects on malaria transmission of treated and untreated nets on the

wider community of bednet users and non-users are still little understood, despite some

recent progresses (Killeen et al., 2007). Randomised trials in different malaria transmission

regions examined the effect of ITNs on mortality of children without bednets. A study

carried out in northern Ghana estimated that mortality risk in individuals without insec-

ticide nets increased by 6.7% with every 100 m shift away from the nearest intervention

compound (Binka et al., 1998). In western Kenya households without ITNs but within

300 m of ITN villages received nearly full protection (Hawley et al., 2003). These results

conflict with those found from studies in The Gambia which concluded that protection

against malaria seen in children using ITN is due to personal rather than community effect

(Lindsay et al., 1993; Thomson et al., 1995, Quinones et al., 1998). A better understand-

ing of these spatial effects in real-life situations is paramount for setting control targets,

especially for understanding equity issues since these spatial effects mainly improve the

situation of unprotected individuals, who are on average poorer. Moreover, the spatial

effects of ITNs on non-bednet users in relation with the degree of density of bednets will

indicate the type and level of bednet coverage that control programs need to achieve in

order to maximize protection of non-bednet users. Here we present for the first time results

for the spatial effects of ITNs in a ”real-life” programme. One of the limitations of pre-

vious studies is that they used standard statistical methods which assume independence

between observations. When these methods are applied to spatially correlated data, they

underestimate the standard errors and thus overestimate the statistical significance of the

covariates (Ver Hoef et al., 2001). In this paper we analyzed data from a large follow-up

study in a highly malaria endemic area in southern Tanzania. Making use of a demo-

graphic surveillance system (DSS) we tracked child mortality prospectively and assessed


the relation between all-cause child mortality rates and the spatial effect of bednet density.

To account for spatial clustering we fitted Bayesian geostatistical models with household-

specific random effects. Models for geostatistical data introduce the spatial correlation

in the covariance matrix of the household-specific random effects and model fit is based

on Markov chain Monte Carlo methods (MCMC). MCMC estimation requires repeated

inversions of the covariance matrix which, for large number of locations is computationally

intensive and time consuming. To address this problem we propose a convolution model

for the underlying spatial process which replaces large matrix inversion by the inversion of

much smaller matrices.

2.2 Methods

2.2.1 Study area and population

The study was carried out from October 2001 to September 2003 in the 25 villages covered

by a demographic surveillance system (DSS) in the Kilombero Valley, southern Tanzania.

The DSS updates every 4 months demographic information on a population of about 73, 000

people living in 12, 000 dispersed households (Figure 2.1) in two districts - Kilombero and

Ulanga (Armstrong Schellenberg et al., 2002). Most residents practice subsistence farming

with rice and maize being the predominant crops. The climate is marked by a rainy season

from November to May with annual rainfall ranging from 1200 to 1800 mm. Malaria is the

foremost health problem, for both adults and children (Tanner et al., 1991). The prevailing

malaria vectors in this region are Anopheles gambiae and Anopheles funestus with an

estimated average entomological inoculation rate estimated of over 360 infective bites per

person a year (Killeen et al., unpublished data). A large-scale social marketing programme

of ITNs for malaria control has been running in this area since 1997 (Schellenberg et al.,

2001).

2.2 Methods 21

Figure 2.1: Distribution of the DSS households according to their socio-economic statusand of the health facilities.

2.2.2 Data collection

Mortality data were obtained prospectively and continuously over a two-year period from

the DSS, which allowed us to register age and sex data, births and migrations in and out

the study area. Exact procedures are described in (Armstrong Schellenberg et al., 2002).

An additional survey was carried out in the DSS population in 2002 to collect socio-

economic information. The survey questionnaire included a list of household assets (e.g.

bednet), housing characteristics (e.g. type of roofing material) and type of energy and

light. Although information on ITNs ownership was also collected, we did not use these

data in our analysis since it was shown (Erlanger et al., 2004) that in this area two-thirds

of the nets that were reported as having been re-treated within the last 12 months had

insufficient insecticide to be effective.

Households and health facilities were geolocated using a hand-held Global Positioning


System (Garmin GPS 12, Garmin corp.) and Euclidean distances between houses and the

health facilities were calculated.

2.2.3 Statistical analysis

Bednet density was defined as the number of bednets per person within a certain radius

around each household. The following radii were chosen: 0 m (bednet coverage at household

level), 50 m, 100 m, 150 m, 200 m, 300 m, 400 m, 500 m and 600 m.

A wealth index was calculated as a weighted sum of household assets. It has been shown

that there is an inverse relationship between mortality and socio-economic status (Gwatkin,

2005); therefore the weights of the wealth index were obtained from the coefficients of a

negative binomial model which estimated the effect of assets on all-age mortality. The

weight of asset i was calculated as wi = bi√∑i b2i

, where bi is the regression coefficient

corresponding to asset i . The wealth index was divided into quintiles corresponding to

poorest, very poor, poor, less poor and least poor groups of the population.

Negative binomial models were fitted to assess the effect of different bednet density mea-

sures on child mortality after adjusting for possible confounders: sex, wealth index and

distance to the nearest health facility, using STATA v. 9.0 (Stata Corporation, College

Station, TX, USA).

To estimate the effect of bednet density on the mortality of children without nets we

performed a similar analysis. In particular, we defined bednet density as above, considering

as index households the ones without any bednet. We then fitted the negative binomial

models adjusted for the above mentioned confounders.

The household mortality data are correlated in space since common environmental risk fac-

tors, proximity to breeding sites and socio-economic exposures may influence the mortality

outcome similarly in households within the same geographical area. The independence

assumption of the standard negative binomial models may result in overestimation of the

significance of the bednet coverage covariate. To address this problem Bayesian geosta-

tistical negative binomial models were fitted with household-level random effects. Spatial

correlation was modeled by assuming that the random effects are distributed according to

a multivariate normal distribution with variance-covariance matrix related to an exponen-

tial correlation function between household locations, i.e. σ2exp(−dijρ), where dij is the

2.3 Results 23

Euclidean distance between households i and j, σ2 is the geographic variability known as

the sill and ρ is the rate of correlation decay. The distribution of random effect defines

the so called Gaussian spatial process. Model fit requires the inversion of a covariance

matrix with the same size as the sample size. Due to the large number of observations

in our dataset, the estimation of model parameters becomes unstable and unfeasible. To

overcome this problem we propose a model based on a convolution representation that

is, we approximate the spatial random process by a weighted sum of a small number of

stationary spatial processes. The size of the covariance matrix that needs to be inverted

is then much smaller, therefore the method is computationally efficient. We employed

Markov chain Monte Carlo simulation to estimate the model parameters. Further details

on this modeling approach are given in the appendix. The analysis was implemented using

software written by the authors in FORTRAN 95 (Compaq Visual FORTRAN Professional

6.6.0) using standard numerical libraries (NAG, The Numerical Algorithm Group Ltd.).

2.3 Results

A total number of 11, 134 children from 7, 403 households had information available on

both geolocation and socio-economic covariates.

The pooled data revealed an overall all-age crude mortality rate of 9.5 per 1000 person-years

and an overall child mortality of 26.2 per 1000 person-years with no difference between the

two districts (P = 0.98 and P = 0.73 , respectively).

The insecticide treatment status of the nets was difficult to ascertain, therefore the results

reported in this section refer to bednets only, whether treated or not. The mean bednet

density in Kilombero Valley was 270 nets per 1000 inhabitants. 10, 160 households (85%)

had at least one bednet and the mean number of bednets per household was 1.64.

Table 2.1 shows the overall child mortality rates together with district-specific child mor-

tality rates by sex, socio-economic status, distance to the nearest health facility and bednet

density at household level. Since there were no significant differences between child mor-

tality rates in Kilombero and Ulanga Districts, all further analysis was done by pooling

the data of the two districts. Males had a slightly lower mortality rate than females, but

sex was not significantly associated with childhood mortality rates (Incidence-Rate Ratios

(IRR) = 0.90, P = 0.22). Similarly, socio-economic status was not significantly associated


with child mortality (P = 0.12), but we could notice a trend for children from the relatively

better off households to have a lower mortality rate than their poorer counterparts. No

significant association was observed with distance to the nearest health facility, but chil-

dren living ≥ 1 km away from the nearest health facility tended to have higher mortality

rates than those living in close proximity.

Explanatory variables Number of children(%) Child mortality ratea P -value

Overall Kilombero Ulanga

Sex

Female 5669 (50.9) 27.6 29.8 25.0 0.81

Male 5465 (49.1) 24.7 27.1 21.6 0.80

Socio-economic status

Poorest 2203 (19.8) 31.1 36.5 26.0 0.75

Very poor 2265 (20.3) 26.0 27.5 24.1 0.92

Poor 2281 (20.5) 25.7 29.5 20.6 0.79

Less poor 2239 (20.1) 21.3 21.1 21.6 0.99

Least poor 2146 (19.3) 27.1 28.9 24.5 0.90

Distance to nearesthealth facility

< 1 km 2793 (25.1) 23.3 25.9 20.7 0.86

1− 4.9 km 4666 (41.9) 27.2 29.4 24.1 0.82

≥ 5 km 3675 (33.0) 26.9 29.0 24.7 0.87

Bednet density athousehold level b

0 1199 (10.8) 28.9 40.4 19.5 0.66

0− 0.2 2531 (22.7) 27.9 28.9 26.8 0.95

0.2− 0.3 3426 (30.8) 28.5 30.3 28.5 0.87

0.3− 0.5 3026 (27.2) 22.3 22.4 22.3 0.99

> 0.5 952 (8.5) 22.6 28.9 13.9 0.79

a : Mortality rate per 1000 person years.b : Number of bednets per person within a 0 m radius around each household.

Table 2.1: Overall and district-specific child mortality rates by sex, socio-economic status,distance to the nearest health facility and bednet density at household level

2.3 Results 25

A simple bivariate analysis showed that bednet density at household level was significantly

associated with reduced child mortality (IRR = 0.50, P = 0.02). There was a tendency

for mortality rates to decrease for children living in households with at least 30% bednet

density coverage.

The effect of various bednet density measures on child mortality after adjusting for possible

confounders is shown in Table 2.2. Surprisingly, the only measure significantly associated

with child mortality was the bednet density at household level (R0) (IRR = 0.53, P =

0.04). We noted that the mean bednet density was similar for all radii, whereas the

standard deviation tended to become smaller as the radius was increasing.

Bednet density Mean (St.dev.) % of households IRRa 95% CI LRTb P -valuec

without bednetsR0 0.25 (0.15) 0.00 0.53 (0.29,0.97) 4.37 0.04

R50 0.18 (0.20) 0.09 0.64 (0.40,1.03) 3.48 0.06

R100 0.24 (0.18) 12.68 1.13 (0.73,1.74) 0.27 0.61

R150 0.25 (0.13) 13.84 1.18 (0.61,2.30) 0.24 0.62

R200 0.26 (0.12) 14.44 1.69 (0.79,3.61) 1.79 0.18

R300 0.26 (0.09) 15.04 2.51 (0.96,6.55) 3.46 0.06

R400 0.27 (0.08) 15.04 2.10 (0.79,5.59) 2.05 0.15

R500 0.27 (0.07) 15.18 2.40 (0.70,8.25) 1.89 0.17

R600 0.27 (0.07) 15.23 2.89 (0.74,11.25) 2.32 0.13

a : IRR:Incidence-rate ratios.b : LRT:Likelihood ratio test.c : P -value based on likelihood ratio test (LRT).

Table 2.2: Summary of bednet density measures and estimates of the effect of bednetmeasures on child mortality, adjusted by sex, socio-economic status and distance to thenearest health facility. Results obtained by fitting negative binomial models.

The results of the bivariate and multivariate non-spatial negative binomial models are

shown in Table 2.3. None of the explanatory variables were significantly associated with

child mortality, except the fourth wealth quintile. After taking into account the spatial cor-

relation present in the data, the effect of the covariates remained non-significant. However,

the confidence intervals became wider, confirming the importance of taking into account

spatial correlation when analyzing geographical data (Cressie, 1993). The parameters σ2


and ρ shown in Table 2.3 measure the spatial variance and the rate of correlation decay

(smoothing parameter), respectively. The estimates of the smoothing parameter ρ indicate

a low spatial correlation in the child mortality rate data. In fact ρ was estimated to be

6.97 km, which in our exponential setting is translated to a minimum distance for which

spatial correlation decrease to 0.05 of only around 0.43 km.

Indicator Bivariate model Multivariate model Spatial modelIRRa 95% CI IRRa 95% CI IRRa 95% CI

Sex

Female 1.0 1.0 1.0

Male 0.90 (0.75,1.07) 0.89 (0.75,1.06) 0.88 (0.73,1.06)

Socio-economic status

Most poor 1.0 1.0 1.0

Very poor 0.83 (0.64,1.09) 0.84 (0.65,1.11) 0.87 (0.63,1.21)

Poor 0.82 (0.63,1.08) 0.84 (0.64,1.10) 0.82 (0.64,1.05)

Less poor 0.69 (0.52,0.91) 0.70 (0.53,0.93) 0.68 (0.51,0.94)

Least poor 0.87 (0.67,1.14) 0.90 (0.69,1.18) 0.90 (0.68,1.20)

Bednet densityat household level

0 1.0 1.0 1.0

0− 0.2 0.96 (0.71,1.32) 0.99 (0.73,1.36) 1.03 (0.83,1.29)

0.2− 0.3 0.99 (0.73,1.33) 1.02 (0.76,1.39) 1.04 (0.81,1.39)

0.3− 0.5 0.77 (0.57,1.06) 0.81 (0.59,1.11) 0.84 (0.56,1.13)

> 0.5 0.78 (0.52,1.17) 0.81 (0.54,1.23) 0.76 (0.54,1.24)

Distance to nearesthealth facility 0.80 (0.16,3.96) 0.60 (0.10,3.68) 0.23 (0.03,3.71)

Spatial parameters

σ2 0.75 (0.35,1.16)

Range (3/ρ)b 0.43 (0.39,0.48)

a : IRR:Incidence-rate ratios.b : Spatial correlation is significant (> 5%) within this distance.

Table 2.3: Results of the association of sex, socio-economic status, bednet density athousehold level and distance to nearest health facility with child mortality, resulting fromthe bivariate and multivariate non-spatial models and spatial model.

2.4 Discussion 27

Table 2.4 depicts the effect of different bednet density measures on the mortality of children

without any bednet after adjusting for sex, socio-economic status and distance to the

nearest facility. The results show no significant association between any bednet density

measure and mortality of children without nets, indicating no detectable community effect.

Bednet Incidence risk ratio P -valuea

density No bednet 0− 0.2 0.2− 0.3 > 0.3R50 1.0 0.88 (0.42,1.83) 0.70 (0.31,1.60) 0.89 (0.42,1.71) 0.94

R100 1.0 0.64 (0.29,1.42) 1.04 (0.51,2.10) 1.15 (0.57,2.34) 0.52

R150 1.0 0.82 (0.34,1.98) 0.91 (0.38,2.17) 2.06 (0.91,4.64) 0.08

R200 1.0 0.74 (0.30,1.79) 0.68 (0.28,1.63) 1.35 (0.57,3.20) 0.30

R300 1.0 1.18 (0.33,4.16) 1.64 (0.48,5.62) 1.40 (0.38,5.08) 0.71

R400 1.0 1.40 (0.31,6.28) 1.90 (0.44,8.24) 1.49 (0.32,7.00) 0.81

R500 1.0 1.97 (0.26,15.15) 2.31 (0.31,17.38) 1.80 (0.22,14.54) 0.73

R600 1.0 1.22 (0.16,9.33) 1.63 (0.22,12.03) 1.14 (0.14,9.06) 0.81

a : P -value based on likelihood ratio test (LRT).

Table 2.4: Estimated effect of bednet measures on mortality of children without nets,adjusted by sex, socio-economic status and distance to the nearest health facility, obtainedby fitting negative binomial models.

Pearsons correlation coefficient between bednet density and bednet usage was 0.83, indi-

cating a strong correlation between the two measures. Hence, the results regarding the

bednet density could be extended to bednet usage.

2.4 Discussion

We examined the effect of a variety of factors on child mortality in an area of high perennial

malaria transmission in southern Tanzania and identified that the density of household bed

net ownership was the only factor significantly associated with child mortality reduction.

The spatial effects of bednets on all-cause child mortality in an area of high perennial

malaria transmission in southern Tanzania have been presented here. The effect of different

bednet density measures was estimated after adjusting for possible confounders like sex,

socio-economic status and distance to the nearest health facility. We concentrated on

all-cause child mortality because in rural Africa it is difficult to assess malaria-specific


mortality. Most deaths occur at home and verbal autopsy is the only tool available to

determine the cause of mortality. It has been shown (Snow et al., 1992; Todd et al., 1994)

that this is an inaccurate method to detect malaria, having a low sensitivity and specificity.

Our results indicated a surprising lack of community effect of bednets on childhood mor-

tality. This conclusion is based on the fact that only the bednet density at household level

had a significant protective effect on child mortality. When net density within ≥ 50m was

considered, the risk of child mortality increased slightly but the relation was not signifi-

cant. Our findings contrast with previous studies in Africa, which demonstrated a strong

community-wide effect of ITNs on child mortality (Binka et al., 1998, Hawley et al., 2003).

However, our study differed from the studies mentioned above in a number of ways.

Firstly, the epidemiological studies that demonstrated the mass effect of ITNs on child

mortality were all designed as community-trial interventions, ensuring a uniformly high

coverage of treated nets in the intervention group, with a control group almost not using

any sort of nets. This creates a strong gradient of ITN at the margins use, which allows a

good measure of spatial effects. By contrast, net usage, treated or not, was uniformly high

in our study area, with the result that any sort of spatial effects would be more difficult to

detect unless there would be heterogeneity in coverage, which was not the case.

Secondly, we were not able to distinguish between treated and untreated nets in the field

because there is no reliable testing method to do this at present. Armstrong-Schellenberg

et al. (2002b) and Erlanger et al. (2004) showed that in our study area use of insecticide

re-treatment is relatively low, with only 32% of the nets having enough insecticide to

ensure an entomological impact. Since untreated nets are less effective than treated ones

(Lengeler, 2004; Maxwell et al., 1999; D’Alessandro et al., 1995b), this had certainly an

impact on the analysis by reducing differences between users and non-users.

Lastly, as specific data on bednet use was not available for the whole sample, we created

a different measure of the impact of bednets: the ”bednet density” defined as the ratio

between the number of bednets owned and the number of people living in a specific area.

Previous studies in this region showed that on average 2 people sleep under a bednet with

an overall bednet use of about 75% (Killeen et al., unpublished data).

Most analyzes of bednets effect on different malaria-related outcomes so far have been based

on the assumption of independence between observations. However, household mortality

data are spatially correlated due to common exposures. When the spatial correlation

2.4 Discussion 29

present in the data is ignored, the statistical significance of the covariates is overestimated.

We could control for that by using a Bayesian geostatistical approach to assess the child

mortality-bednet density relation. Bayesian computation implemented via MCMC enabled

simultaneous estimation of all model parameters together with their standard errors, a

feature that is not available in the maximum likelihood based framework.

Despite these limitations, our results are consistent with the analysis of ITNs protective

efficacy against malaria transmission in Kilombero Valley (Killeen et al., 2006), which

predicted little community-level protection for the individuals not using ITNs. The most

likely explanations for this were the small proportion of re-treated nets and the insufficient

concentration of insecticide present in the bednets, leading to diversion of mosquitoes. A

recently developed model for the transmission of malaria using data collected in Tanzania

(Killeen et al., 2007) predicted that modest bednet coverage (35% - 65%) of the entire pop-

ulation, rather than just high-risk groups (pregnant women and young children) is needed

to achieve community-wide protection similar to, or greater than, individual protection.

Hence, there is clearly a strong case for improving the status of insecticide treatment

through the introduction of long-lasting insecticidal nets (LLINs) which are now becoming

increasingly available (Guillet et al., 2001) and for the wide-use of ITNs and LLINs by

the whole population. We expect that achieving a high coverage with LLINs will result in

further substantial reductions of malaria transmission and hence malaria-related mortality

and morbidity for both users and non-users.

Acknowledgments

The authors would like to acknowledge the residents of the Kilombero Valley for their

commitment during the study and the Ifakara DSS field and office workers who carried

out the surveys and compiled the database. We would like to thank Prof. Tom Smith and

Prof. Dr. Don de Savigny for stimulating discussions. Many thanks also to Dr. Hassan

Mshinda, Dr. Honorati Masanja, Oskar Mukasa and Dr. Salim Abdulla for their overall

support. This manuscript has been published with kind permission of Dr Andrew Kitua,

Director of the National Medical Research Institute, Tanzania. This study was funded by

the Swiss National Science Foundation (Grant number 3252B0-102136/1). Ethical review

and approval for this study was provided by the Medical Research Coordination Committee

of NIMR (Ref. No NIMR/HQ/R.8a/VOL.X/12, dated 28/4/1998).


2.5 Appendix

Let Yil be the mortality outcome of child l at site si i = 1, . . . , n taking value 1 if the child

is dead and 0 otherwise. We assume that Yil arises from a negative binomial distribution,

that is Yil ∼ NegBin(pil, r), where pil is the probability that child l at location si is

dead and r is the parameter that quantifies the amount of extra Poisson variation. To

account for spatial variation in the data, location-specific random effects were integrated

in the negative binomial model. The probability pil is modeled as pil = rr+zil

, with log(zil) =

log(pyrsil)+XTil β+φi, where Xil is the vector of associated covariates, β are the regression

coefficients and φi’s are the spatial random effects. pyrsil represents the number of person-

years corresponding to child l at location si and log(pyrsil) is considered as covariate with

regression coefficient fixed to 1 and is referred as offset.

The standard approach to model the spatial dependence is to assume that the covariance

of φi’s at every two locations si and sj decreases with their distance dij, that is Σij =

σ2f(dij; ρ) with f(dij; ρ) = exp(−dijρ), where ρ > 0 is a smoothing parameter that controls

the rate of correlation decay with increasing distance and σ2 quantifies the amount of

spatially structured variation. Estimation of the location-specific random effects and of

the spatial parameters requires repeated inversions of the covariance matrix Σ. Due to

the large number of locations in our dataset (7, 403), matrix inversion is computationally

intensive and is not feasible within practical time constrains. To overcome this issue

we develop a convolution model for the underlying spatial process. In particular, we

choose a small number of locations tk, k = 1, . . . , K over the study region, assume a

stationary spatial process ωk over these locations and we model the spatial random effect

φi at each data location si as a weighted sum of the fixed location stationary processes.

That is, φi =∑K

k=1 a(i, k)ωk, where the weights a(i, k) are decreasing functions of the

distance between data location si and the fixed location tk and ωk ∼ N(0, Σk), with

(Σk)hl = σ2exp(−dhlρ), where dhl is the distance between the fixed locations th and tl.

This approach avoids the inversion of the large covariance matrix nxn , reducing the

problem to the inversion of a much smaller size matrix KxK. For this specific analysis we

have chosen K = 200.

For the correlation function chosen, the minimum distance for which spatial correlation

between locations is below 5% is 3/ρ (range). The above specification of spatial correlation

is isotropic, assuming that correlation is the same in all directions.

2.5 Appendix 31

Following a Bayesian model specification, we adopt prior distributions for the model param-

eters as follows: non-informative uniform prior distributions for the regression coefficients

β, inverse gamma prior distribution for σ2 and gamma prior distribution for the decay

parameter ρ and the over-dispersion parameter r.

We estimate the model parameters using Markov chain Monte Carlo simulation. In partic-

ular we implemented Gibbs sampler (Gelfand and Smith, 1990), which requires simulating

from the full conditional distributions of all parameters iteratively until convergence. The

full conditional distribution of σ2 is an inverse gamma distribution and it is straightfor-

ward to simulate from. The conditional posterior distribution of β, ρ, and r do not have

known forms. We simulate from these distributions using the Metropolis algorithm with a

Normal proposal distribution having the mean equal to the parameter estimate from the

previous Gibbs iteration and the variance equal to a fixed number, iteratively adapted to

optimize the acceptance rates. We have run a five-chain sampler with a burn-in of 10, 000

iterations and we assessed the convergence by inspection of ergodic averages of selected

model parameters after 200, 000 iterations.

Chapter 3

Bayesian modeling of geostatistical

malaria risk data

Gosoniu L.1 ,Vounatsou P.1, Sogoba N.2 and Smith T.1

1 Swiss Tropical Institute, Basel, Switzerland2 Malaria Research and Training Center, Universite du Mali, Bamako, Mali

This paper has been published in Geospatial Health 1, 2006, 127-139

34 Chapter 3. Bayesian modeling of geostatistical malaria risk data

Summary

Bayesian geostatistical models applied to malaria risk data quantify the environment-

disease relations, identify significant environmental predictors of malaria transmission and

provide model-based predictions of malaria risk together with their precision. These mo-

dels are often based on the stationarity assumption which implies that spatial correlation

is a function of distance between locations and independent of location. We relax this as-

sumption and analyze malaria survey data in Mali using a Bayesian non-stationary model.

Model fit and predictions are based on Markov chain Monte Carlo simulation methods.

Model validation compares the predictive ability of the non-stationary model with the sta-

tionary analogue. Results indicate that the stationarity assumption is important because

it influences the significance of environmental factors and the corresponding malaria risk

maps.

Keywords : Bayesian inference; malaria risk; Markov chain Monte Carlo; non-stationarity;

kriging

3.1 Introduction 35

3.1 Introduction

Malaria is the most prevalent human parasitic disease. Although reliable estimates are not

available, rough calculations suggest that globally, 250 million new cases occur each year

resulting in more than one million deaths (Bruce-Chwatt, 1952; Greenwood, 1990; WHO,

2004). Around 90% of these deaths happen in sub-saharan Africa, mostly in children

less than 5 years old. The malaria parasite is transmitted from human to human via the

bite of infected female Anopheles mosquitoes. Transmission depends on the distribution

and abundance of the mosquitoes which are sensitive to environmental factors mainly

temperature, rainfall and humidity. By determining the relations between the disease

and the environment, the burden of malaria can be estimated at places where data on

transmission are not available and high risk areas can be identified. Reliable maps of

malaria transmission can guide intervention strategies and thus optimize the use of limited

human and financial resources to areas of most need. In addition, early warning systems

can be developed to predict epidemics from environmental changes.

Remote sensing is a useful source of satellite-derived environmental data. Geographic

Information Systems (GIS) has emerged over the last 15 years as a powerful tool for linking

and displaying information from many different sources such as environmental and disease

data, in a spatial context. Integrated GIS and remote sensing have been applied to map

malaria risk in Africa (Snow et al., 1996; Craig et al., 1999; Thomson et al., 1999; Hay et al.,

2000; Kleinschmidt et al., 2001; Rogers et al., 2002). However, the mapping capabilities of

existing GIS software are rather limited as they are unable to quantify the relation between

environmental factors and malaria risk and to produce model-based predictions. GIS is

also used in early warning systems for malaria epidemics (Abeku et al., 2004; Grover et al.,

2005; Thomson et al., 2006), however the thresholds for environmental factors have been

based on expert opinion rather than observed data.

Statistical modeling gives mathematical descriptions of the environment-disease relations,

identifies significant environmental predictors of malaria transmission and provides pre-

dictions of malaria risk based on the above relations together with their precision. The

standard statistical models assume independence of observations. However, malaria infec-

tious cases cluster due to underlying common environments. When spatially correlated

data are analyzed this independence assumption leads to overestimation of the statisti-

cal significance of the covariates (Cressie, 1993). Spatial models incorporate the spatial

correlation according to the way the geographical information is available. For areal data


(typically counts or rates aggregated over a particular set of contiguous units) the spatial

correlation is defined by a neighborhood structure. For geostatistical data (collected at

fixed locations over a continuous study region) the spatial correlation is usually considered

as a function of the distance between locations.

Linear regression is applied for modeling geostatistical continuous data which are normally

distributed (Gaussian). The spatial correlation is introduced in the residuals (error terms)

of the model. The parameters can not be estimated simultaneously, thus iterative methods

are used. The generalized least squares approach (GLS) estimates the regression coeffi-

cients conditional on the spatial correlation parameters. The correlation parameters can

be estimated conditional on the regression coefficients empirically from the residuals or

using maximum likelihood based approaches (Zimmerman and Zimmerman, 1991).

In this paper we present models for geostatistical prevalence data derived from malaria

surveys carried out at a number of fixed locations. For this type of data and in general

for non-Gaussian geographical data, spatial models introduce at each location an error

term (random effect) and incorporate spatial correlation on these parameters. Estimation

can use generalized linear mixed models (GLMM). However, this is difficult to apply for

spatial problems with large number of locations (Gemperli and Vounatsou, 2004). In

addition, estimation of standard errors depends on asymptotic results, which in the case

of geostatistical models, do not give unique estimates (Tubila, 1975).

Bayesian geostatistical models implemented via Monte Carlo methods avoid asymptotic

inference and the computational problems encountered in likelihood-based fitting. They

were introduced for the analysis of geostatistical data by Diggle et al. (1998) and have

been employed in modeling the spatial distribution of parasitic diseases (Diggle et al.,

2002; Gemperli et al., 2004; Raso et al., 2004; Abdulla et al., 2005; Gemperli et al.,

2005; Raso et al., 2005; Clements et al., 2006; Gemperli et al., 2006; Raso et al., 2006).

Most health applications of Bayesian geostatistical models have relied on an assumption of

stationarity, which implies that the spatial correlation is a function of the distance between

locations and independent of locations themselves. This assumption is questionable when

malariological indices are modeled since local characteristics related to human activities,

land use, environment and vector ecology influence spatial correlation differently at the

different locations.

In this paper we present and compare Bayesian stationary and non-stationary models for

mapping malaria risk data in Mali. Using model validation we assess the assumption of

3.2 Data 37

stationarity and show the impact it can have on inference when non-stationary data are

analyzed. In Section 3.2 we describe the malaria data which motivated this work and the

environmental predictors we extracted from remote sensing and GIS databases. Section

3.3 introduces the stationary and non-stationary Bayesian geostatistical models as well as

the model validation approaches. The results are presented in Section 3.4 and the paper

ends with final remarks and suggestions for future work given in Section 3.5.

3.2 Data

3.2.1 Malaria data

The malaria data were extracted from the ”Mapping Malaria Risk in Africa” (MARA/

ARMA,1998) database. This is the most comprehensive database on malariological indices

initiated to provide a malaria risk atlas by collecting published and unpublished data from

over 10, 000 surveys across Africa. We analyzed malaria prevalence data from surveys

carried out in children between 1 and 10 years old at 86 sites in Mali (Figure 3.1) between

1977 and 1995, including a total of 43, 492 children.

3.2.2 Climatic and environmental data

The environmental data and the databases from which they were extracted are given

in Table 3.1. Preliminary non-spatial analysis indicated that the following factors and

their transformation should be included in the analysis: Normalized Difference Vegetation

Index (NDVI), NDVI squared, length of malaria season, amount of rainfall, maximum

temperature, squared maximum temperature, minimum temperature, squared minimum

temperature, distance to the nearest water body and squared distance to the nearest water

body.

Factor Resolution SourceNDVI 8km2 NASA AVHRR Land data setsTemperature 5km2 Hutchinson et al. 1996Rainfall 5km2 Hutchinson et al. 1996Water bodies 1km2 World Resources Institute 1995Season length 5km2 Gemperli et al. 2006

Table 3.1: Spatial databases used in the analysis.


Figure 3.1: Sampling locations with dot shading indicating the observed malaria preva-lence. The stars indicate the centroids of two fixed tiles used to account for non-stationarity.

The NDVI values were extracted from satellite information conducted by the NOAAA/

NASA Pathfinder AVHRR Land Project (Agbu and James, 1994). NDVI is shown to be

highly correlated with other measures of vegetation (Justice et al., 1985) and used as a

proxy of vegetation and soil wetness. Index values can vary from -1 to 1 with higher values

(0.3−0.6) indicating the presence of green vegetation, and negative values indicating water.

We used the logarithm of the yearly mean NDVI over the malaria season. The temperature

and rainfall data were obtained from the ”Topographic and Climate Data Base for Africa

(1920-1980)” Version 1.1 by Hutchinson et al. (1996). We used the yearly averages over

the months suitable for transmission according to the map of Gemperli et al. (2006).

The distance to the nearest water source was calculated based on permanent rivers and

lakes extracted from ”African Data Sampler” (WRI, 1995). The length of malaria season

was defined using the seasonality model of (Gemperli et al., 2006). The covariates were

standardized prior to the analysis.

3.3 Bayesian geostatistical models 39

3.3 Bayesian geostatistical models

3.3.1 Model formulation

The malaria data are derived from surveys carried out at the various locations. These are

typical binomial data and modeled via logistic regression. Let Ni be the number of children

tested at location si, i = 1, . . . , n, Yi be the number of those found with malaria parasites in

a blood sample and Xi = (Xi1, Xi2, . . . , Xip)T be the vector of p associated environmental

predictors observed at location si. We assume that Yi arises from a Binomial distribution,

that is Yi ∼ Bin(Ni, pi) with parameter pi measuring malaria risk at location si and model

the relation between the malaria risk and environmental covariates Xi via the logistic

regression logit(pi) = XTi β, where β = (β1, β2, . . . , βp)

T are the regression coefficients.

This model assumes independence between the surveys. However, the geographical location

introduces correlation since the malaria risk at nearby locations is influenced by similar

environmental factors and therefore it is expected that the closer the locations the similar

the way malaria risk varies. To account for spatial variation in the data we introduce an

error term (random effect) φi at each location si, that is logit(pi) = XTi β + φi and model

the spatial correlation on the φi parameters, that is the φi’s are not independent but

they derive from a distribution which models the correlation or equivalently the covariance

between every pair of random effects. We adopt the multivariate Normal distribution for

the φi’s since they represent error terms and therefore they are defined on a continuous

scale, that is φ = (φ1, φ2, . . . , φn)T ∼ N(0, Σ). Σ is a matrix with elements Σij quantifying

the covariance Cov(φi, φj) between every pair (φi, φj) at locations si and sj respectively.

The distribution of random effect φ defines the so called Gaussian spatial process.

Stationary model

Assuming stationarity, spatial correlation is considered to be a function of distance only

and irrespective of location. Under this assumption, we take Σij = σ2corr(dij; ρ), where

corr is a parametric correlation function of the distance dij between locations si and sj.

Several correlation functions have been suggested by Ecker and Gelfand (1997). In this

application, we choose an exponential correlation function corr(dij; ρ) = exp(−dijρ), where

ρ > 0 measures the rate of decrease of correlation with distance and it is known as the

range parameter of the spatial process. For the correlation function chosen, the minimum

distance for which the correlation becomes less than 5 % is 3/ρ. σ2 measures within location


variation and it is known as the sill of spatial process. The above specification of spatial

correlation is isotropic, assuming that correlation is the same in all directions.

Non-stationary model

The assumption of stationarity is not always justified, especially over large geographical

areas. Differences in agro-ecological zones, health systems and socio-economic indicators

may change geographical correlation differently at various locations. In recent years, non-

stationary specifications are based on piecewise Gaussian processes (Kim et al., 2002,

Gemperli et al., 2003) kernel convolution methods (Higdon et al., 1999; Fuentes et al.

2002) and normalized distance-weighted sums of stationary processes (Banerjee et al.,

2004). In Raso et al. (2005) we extended the Banerjee et al. (2004) model for non-

Gaussian prevalence data to map hookworm risk in the region of Man in Cote d’Ivoire. In

this paper, we use the same approach to analyze the Mali malaria prevalence data.

The study area is partitioned into K subregions, a stationary spatial process ωk is assumed

in each subregion k = 1, . . . , K that is ωk = (ωk1, . . . , ωkn)T ∼ N(0, Σk) and the spatial

random effect φi at each location si is modeled as a weighted sum of the subregion-specific

stationary processes, that is φi =∑K

k=1 aikωki, where aik are decreasing functions of the

distance between location si and the centroid of the subregion k. This is equivalent to say

that φ = (φ1, φ2, . . . , φn)T ∼ N(0,∑K

k=1 AkΣkAk), where Ak = diag{a1k, a2k, . . . , ank} is

a matrix which has the elements a1k, a2k, . . . , ank on the main diagonal and 0 outside the

main diagonal. The Σk are specified using exponential correlation functions as in the case

of the stationary model described in the previous section, that is (Σk)ij = σ2kexp(−dijρk).

Note that the spatial parameters σk and ρk are specific for each subregion k.

Three non-stationary models were fitted with K = 2, 3, 4. Due to relatively small number

of locations included in our data we have not investigated models with larger number of

tiles to avoid estimating spatial parameters from tiles with few locations and thus over-

parametrising the models. The sub-regions were obtained by overlaying a rectangular grid

over the study area. We first divide the rectangle in half north-to-south and then, to obtain

four sub-regions, each of these rectangles is partitioned in half west-to-east. For K = 3 we

divide the north part of our study area in two rectangles and consider the south area as

one sub-region. The centroids of two fixed tiles are shown in Figure 3.1.

3.3 Bayesian geostatistical models 41

3.3.2 Bayesian specification and implementation

The Bayesian approach to inference allows parameter estimation using information coming

from the data via the likelihood function as well as information coming from other sources

prior seen the data (i.e. previous studies, subjective judgments) which is formalized via

prior distributions. Bayes theorem combines the likelihood function and the prior distri-

bution defining a new quantity, known as posterior distribution which forms the basis of

Bayesian inference. Parameters are considered as random and their estimation results not

only in a single value, but in the probabilities of their possible values which are given by

their probability distribution, known as marginal posterior distribution.

To complete the Bayesian model formulation of the geostatistical models mentioned above

we need to specify prior distributions for their parameters. For the regression coefficients

we adopt a non-informative uniform prior distributions with bounds −∞ and ∞ which

reflects lack of prior knowledge other than that the regression coefficients can take any

positive or negative value. For the spatial parameters σ2, ρ, σ2k and ρk we adopt inverse

gamma and gamma prior distributions respectively with parameters chosen to have mean

equal to 1 and very large variance.

We estimate the parameters of the model using Markov chain Monte Carlo (MCMC)

simulation and in particular Gibbs sampling (Gelfand and Smith, 1990). Starting with

some initial values about the parameters, the algorithm iteratively updates the parameters

by simulating from their full conditional distributions, that is the posterior distribution of

each parameter conditional on the remaining parameters. The full conditional distributions

of σ2 and σ2k, k = 1, . . . , K are inverse gamma distributions and simulation from them is

straightforward. The rest of the parameters do not have full conditional distributions of

known forms. We simulate from the non-standard distributions by employing a random

walk Metropolis algorithm (Tierney, 1994), having a Normal proposal density with mean

equal to the estimate of the corresponding parameter from the previous Gibbs iteration and

variance equal to a fixed number, iteratively adapted to optimize the acceptance rates. We

run five chains with a burn-in of 5, 000 iterations. Convergence was assessed by inspection

of ergodic averages of selected model parameters.

The analysis was implemented in Fortan 95 (Compaq Visual Fortran Professional 6.6.0)

using standard numerical libraries (NAG, The Numerical Algorithms Group Ltd.).


3.3.3 Prediction model

Bayesian kriging (Diggle et al., 1998) is used to predict the malaria risk at locations where

malaria data are not available. This approach treats the malaria risk at a new location as

random and calculates its predictive posterior distribution, which provides not only a single

estimate of the risk but a whole range of likely values together with their probabilities to be

the true values at a specific location. This makes it possible to estimate the prediction error,

a substantial advantage over the classical kriging methods. We estimated the predictive

posterior distributions at new locations via simulation. Predictions were made for 28, 000

pixels, covering the whole area of south Mali. Further details are given in the Appendix of

this chapter.

3.3.4 Model validation

In total we fitted 4 models (a stationary and three non-stationary). Model fit was carried

out on a randomly selected subset of our data (training set) including 69 locations. The

remaining dataset of 20 locations was used for validation (testing set).

The goodness-of-fit of each model was assessed using the Deviance Information Criterion

(DIC) ( Spiegelhalter et al., 2002). This quantity considers the fit of the data but penalizes

models that are very complex.

The predictive ability of the models was assessed using a Bayesian ”p-value” analogue

calculated from the predictive posterior distribution. In particular, for each one of the

test locations we calculated the area of the predictive posterior distribution which is more

extreme than the observed data. The model predicts the observed data well for a spe-

cific location when the observed data is close to the median of the predictive posterior

distribution and therefore the ”p-value” close to 0.5. A box plot is used to summarize

the ”p-values” calculated from the 20 test locations under a particular model. The box

plot displays the minimum, the 25th, 50th, 75th quantile as well as the maximum of

the distribution of the 20 ”p-values”. We consider as best the model with median ”p-

value” closer to 0.5. The ”p-value” is calculated using simulation-based inference by1

1000

∑1000j=1 min(I(p

rep(j)i > pobs

i ), I(prep(j)i < pobs

i )), where pobsi is the observed prevalence

at test site si and prepi = (p

rep(1)i , . . . , p

rep(1000)i ) are 1000 replicated data from the predictive

distribution at test location si.

3.4 Results 43

A χ2-based measure was also calculated as an alternative way of comparing the predictive

ability of the models. For every test location si, we calculated the statistic χ2i =

(Y obsi −Yi)

2

Yi

where Y obsi are the observed count at test location si and Yi is the median of the predictive

posterior distribution at si. For each model, we obtained the distribution as well as the

sum Tχ2 of the χ2i values over the 20 test points. The best model was the one with the

lowest median and Tχ2 , estimating predicted counts which are closer to the observed ones.

In addition to the above approaches, for each model we calculated 5 credible intervals (the

equivalent of confidence intervals in the Bayesian framework) with probability coverage

equal to 5%, 25%, 50%, 75% and 95% respectively of the posterior predictive distribution

at the test locations. The model which gave better predictions was the one with the highest

percentage of locations within the interval of smallest coverage.

3.4 Results

The pooled data have shown an overall malaria prevalence of 44.0%(19, 156 children). The

median malaria prevalence estimated at village level was 51.3%, ranging from 5.3% to

95.5%.

The univariate non-spatial analysis showed that the following environmental indicators

and their transformations were associated with malaria prevalence: NDVI, length of ma-

laria season, rainfall, maximum temperature, minimum temperature and distance to the

nearest water body. The relation with NDVI was best described by the logarithmic trans-

formation and the relation with minimum, maximum temperature and distance to water

by polynomial terms of order 2. The results of the bivariate non-spatial logistic regression

are summarized in Table 3.3. All covariates significant at a 15% significance level were

included in the spatial analysis.

Figure 3.2 compares the predictive ability of the stationary and 3 non-stationary (with 2,

3 and 4 tiles respectively) multiple logistic regression models using the Bayesian ”p-value”

approach. Each box plot summarize the distribution of the 20 ”p-values” calculated from

the predictive posterior distribution of the 20 test locations. The median of this distribution

for the non-stationary model with two tiles is the closest to 0.5, suggesting that this is the

best model. The same conclusion was drawn by comparing the models using the chi-

squared measure. Figure 3.3 shows that the non-stationary models with two and three


tiles have similar medians of the distribution of χ2-values over the 20 test locations, but

the non-stationary model with two tiles had the lowest Tχ2 value, indicating the smallest

deviations between the observations and model predictions.

Figure 3.2: The distribution of Bayesian p-values for the stationary model (ST), and thenon-stationary with 2 (NS2), 3 (NS3), and 4 (NS4) tiles.

In Table 3.2 are presented the percentages of test locations with malaria prevalence which

falls in each of the 5 credible intervals of the posterior predictive distribution. We observe

that the non-stationary model with two fixed tiles includes 10% of the test locations in the

narrowest interval of 5% probability content. This is the highest percentage in comparison

to the remaining fitted models. Also in the 95% credible interval the non-stationary model

with two fixed tiles has the highest percentage of observed prevalences at test locations,

namely 80% in comparison with 75% reported by the other three models.

3.4 Results 45

Figure 3.3: The distribution and the sum Tχ2 of the χ2-values over the 20 test points.

Credible Interval Bayesian geostatistical modelStationary NS-2 tiles NS-3 tiles NS-4 tiles

5% 5% 10% 0% 5%

25% 15% 25% 15% 25%

50% 30% 55% 50% 35%

75% 55% 60% 65% 55%

95% 75% 80% 75% 75%

Table 3.2: Percentage of test locations with malaria prevalence falling in the 5%, 25%,50%, 75% and 95% credible intervals of the posterior predictive distribution.

Table 3.3 depicts the results of the stationary and the best fitting non-stationary model

with two tiles. The stationary model suggested that the following environmental factors are

associated with malaria risk: NDVI (in logarithmic scale), maximum, minimum tempera-

ture and distance to the nearest water body (in polynomial forms of order 2) and rainfall.


In the non-stationary model the rainfall as well as the second order polynomial of the

distance to water were not any more related with the malaria risk. As we were expected,

the higher the value of the NDVI (indicating the presence of green vegetation) the higher

the malaria risk. A negative relation with maximum temperature showed that the lower

the maximum temperature the higher the malaria risk. Also, malaria risk increases with

an increase in the minimum temperature. Surprisingly, the models estimated a positive

relation with the distance to water, implying that the risk increases with the distance from

permanent water bodies.

The stationary model calculates a posterior median for ρ equal to 2.63 (95% credible

interval: 1.11, 6.09) which, in our exponential setting indicates that the minimum distance

for which the spatial correlation is smaller than 5 % is equal to 3/ρ = 1.14 km (95%

credible interval: 0.49, 2.71). The best fitting 2-tile non-stationary model confirms that

spatial correlation changes as we move from the North to the South part of the country.

In particular the minimum distance with negligible correlation is 0.86 km (95% credible

interval: 0.40, 2.19) in the North and 8.90 km (95% credible interval: 1.79, 26.88) in the

South part. It is interesting to see that although the models differ in their predictive ability

(Figure 3.2 and Figure 3.3), the goodness of fit DIC measure does not favor any of the

models, showing that it is not able to assess which model has the best predictions.

The smooth maps of malaria prevalence in sub-Sahara Mali obtained from the stationary

and non-stationary spatial model with two tiles are shown in Figures 3.4 and 3.5. Both

maps predicted high malaria prevalence in the region of Kayes (South-West Mali), with

the exception of the district of Kayes and in the region of Segou (East-Center Mali). Low

prevalence was predicted in the regions of Gao, Tombactou and Kidal (North Mali) and in

the district of Kati (Center Mali). Differences between the stationary and non-stationary

models appear in the districts of Ansongo, Gourma Rharous, Douentza and western district

of Tombactou region (Goundam). Figures 3.6 and 3.7 depict the prediction error from the

stationary and non-stationary models respectively. The error is higher in the North Mali

where the observed data were very sparse. The prediction error obtained from the non-

stationary model was lower, ranging from 0.36 to 5.7 in comparison to that obtained from

the stationary one which varied from 0.70 to 8.11.

3.4 Results 47Var

iable

Biv

aria

teSta

tion

ary

Non

-sta

tion

ary

(2tile

s)non

-spat

ialm

odel

spat

ialm

odel

spat

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odel

Med

ian

95%

CIa

Med

ian

95%

CIa

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ian

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CIa

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rcep

t0.

13(-

0.25

,0.

50)

0.21

(-0.

20,0.

63)

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(ND

VI)

0.26

(0.2

4,0.

28)

0.97

(0.4

3,1.

49)

0.85

(0.2

8,1.

40)

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(ND

VI)

2-0

.11

(-0.

12,-0

.10)

0.20

(-0.

15,0.

56)

0.13

(-0.

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47)

Sea

son

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gth

0.24

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30)

Rai

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)-0

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-0.6

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01)

Max

imum

Tem

per

ature

-0.4

0(-

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,-0

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,-0

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-1.0

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Max

imum

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ature

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(-0.

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(-0.

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0.05

(-0.

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Min

imum

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per

ature

-0.0

5(-

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,-0

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0.94

(0.3

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0.90

(0.2

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Min

imum

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per

ature

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(-0.

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01)

-0.3

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tance

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ater

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tance

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ater

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10(0

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(0.5

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ρb

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51)

DIC

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4750

7.50

a:

Cre

dib

lein

terv

als

(or

pos

terior

inte

rval

s).

b:

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eca

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ary

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each

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.

Tab

le3.

3:Pos

teri

ores

tim

ates

for

model

par

amet

ers.


Figure 3.4: Map of predicted malaria risk for southern Mali using the stationary model.

Figure 3.5: Map of predicted malaria risk for southern Mali using the non-stationary modelwith 2 fixed tiles.

3.4 Results 49

Figure 3.6: Map of prediction error for southern Mali using the stationary model.

Figure 3.7: Map of prediction error for southern Mali using the non-stationary model with2 fixed tiles.


3.5 Discussion

Accurate maps of malaria risk are important tools in malaria control as they can guide in-

terventions and assess their effectiveness. These maps rely on predictions of risk at locations

without observed prevalence data. Malaria is an environmental disease and environmen-

tal factors are good predictors of transmission, but the relation between environmental

factors, mosquito abundance and malaria prevalence is not linear. This relation can be

established only by means of adequate spatial statistical models which can be used for

improving predictions of malaria transmission not only in space (for risk mapping) but

also in time (for developing early warning systems for malaria epidemics). In this study

we present Bayesian geostatistical approaches to assess the malaria-environmental relation

for the purpose of malaria risk mapping.

The Bayesian stationary and non-stationary models we presented for analyzing the malaria

survey data in Mali showed that the statistical modeling approach plays an important role

in inference. It influences not only the estimation of parameters related with the spatial

structure of the data but also the significance of the malaria risk predictors, the resulting

malaria risk maps and the associated predicted errors. Model validation should routinely

accompany any model fitting exercise. For the purpose of validation, we recommend to

carry out the model fitting on the 80% of the data locations and compare the predictive

ability of the models on the remaining locations. For the purpose of mapping, we suggest,

once the best model is selected, to apply it to the whole dataset so that the final maps are

based on as much data as possible.

Non-stationarity is an important feature of malaria data which is often ignored. Gemperli

(2003) developed a non-stationary model for analyzing malaria risk data which divides the

study region in random tiles, assuming a separate correlation structure within region but

independence between tiles. The number and configuration of tiles are random parameters

estimated by the data. The non-stationary modeling approach we adopt here relies on a

partition of the study region into fixed tiles. We assume a separate correlation structure

within tile as well as correlation between tiles. This modeling approach is more appropriate

when modeling malaria data over large areas covering different ecological zones which

define the fixed partition. An extension of the model will allow different covariate effects

in each zone. We are currently working on such an approach and implementing it in

analyzing MARA malaria risk data from West and Central Africa. A further extension of

the methodology presented here is to assume random rather than fixed partition of region

3.5 Discussion 51

in tiles. This methodology could be applied in mapping malaria data over large areas with

no clear way of finding a fixed partition (i.e no clearly defined ecological zones).

The main advantage of the Bayesian model formulation is the computational ease in model

fit and prediction compared to classical geostatistical methods. Both the stationary and

especially the non-stationary models have a large number of parameters. Bayesian compu-

tation implemented via MCMC enables simultaneously estimation of all model parameters

together with their standard errors. In addition, Bayesian kriging allows model-based pre-

dictions (together with the prediction error) taking into account the non-stationary feature

of the data. This is not possible in a maximum likelihood based framework.

The significant positive association between our data and the distance to water was un-

expected. Possible explanation could be because the majority of the main cities (most

populated areas) in Mali are located along the river Niger. During the dry season the

receding of the river create numerous water pools which serve as vector breeding habitats.

The time lag between the rainfall and vector abundance and between vector abundance

and the occurrence of the disease may have also played an important role.

Earlier analyzes of the MARA data in Mali (Kleinschmidt et al., 2000; Gemperli, 2003)

differ in the way the spatial structure is incorporated in the model as well as in the way

the covariate effects were modeled. Kleinschmidt et al. (2000) determined the relation

between malaria prevalence and environmental predictors by fitting an ordinary logistic

regression by maximum likelihood method without taking into account spatial correlation.

The prediction map was improved by kriging the residuals and adding them to the map on

a logit scale. The main weaknesses of this analysis are firstly that estimation of environ-

mental effects did not take into account the spatial correlation and thus the significance

of the covariates may have been underestimated; and secondly the kriging assumes nor-

mality, which usually does not hold for the residuals of the logistic regression. Gemperli

(2003) re-analyzed the data using the Bayesian non-stationary model with random tiles

mentioned above. Both previous analyzes found a negative relation between malaria risk

and distance to water, while Gemperli (2003) suggested also a positive relation with rain-

fall. Neither analyzes assessed non-linear covariate effects. The different analyzes reported

different covariate effects and produced different maps of prevalence from essentially the

same database. Neither performed model validation on test data.

The predicted prevalence map from the non-stationary model with 2 tiles is in a better

agreement with the eco-geographical descriptive epidemiology of malaria in Mali (Doumbo


et al., 1989) than the maps obtained from the other models. The two maps of predicted

malaria prevalence obtained from the stationary and the non-stationary model with two

tiles were shown to different malaria epidemiologists in Mali. They all agreed that that

the non-stationary model predicts better the epidemiological situation of malaria in Mali.

However, they found that the prevalence in the western part of the country (Kayes region)

is over-estimated in comparison with the southern region of Mali (Sikasso). Also previous

mapping approaches (Kleinscmidt et al., 2000; Gemperli, 2003) suggested high malaria

prevalence in the western region of Mali. The relatively high predictive standard deviation

observed in the North-Western (region of Kayes) and the desert fringes (Tombouctou, Gao

and Kidal regions) of the country is probably because of the very few number of data points

in these areas rather than the statistical approach. Only one survey has been carried out

in the northern regions since 1988.

Further analyzes which include recent data, particularly in areas where very few number

of data points were observed such as in the north part are needed because environmental

changes in the last decades are likely to have influenced malaria transmission dynamics in

Mali.

Acknowledgments

The authors would like to acknowledge the MARA / ARMA collaboration for making the

malaria prevalence data available. We are thankful to Dr. Sekou F. Traour, Dr. Seydou

Doumbia, Dr. Madama Bouar and Dr. Mahamoudou Tour for their useful comments on

the predicted risk maps. This work was supported by the Swiss National Foundation grant

Nr.3252B0-102136/1.

3.6 Appendix 53

3.6 Appendix

Once the spatial parameters are estimated and the environmental covariates X0 at unsam-

pled locations are known, we can predict the malaria risk at new sites s0 = (s01, s02, . . . , s0l)T

from the predictive distribution

P (Y0|Y , N ) =∫

P (Y0|β, φ0)P (φ0|φ, σ2, ρ)P (β, φ, σ2, ρ|Y , N ) dβ dφ0 dφ dσ2 dρ,

where Y0 = (Y01, Y02, . . . , Y0l)T are the predicted number of cases at locations s0,

P (β, φ, σ2, ρ|Y , N ) is the posterior distribution and φ0 is the vector of random effects at

new site s0. The distribution of φ0 at unsampled locations given φ at observed locations

is normal

P (φ0|φ, σ2, ρ) = N(Σ01Σ−111 φ, Σ00 − Σ01Σ

−111 ΣT

01)

with Σ11 = E(φφT ) the covariance matrix built by including only the sampled locations

s1, s2, . . . , sn, Σ00 = E(φ0φT0 ) the covariance matrix formed by taking only the new lo-

cations s01, s02, . . . , s0l and Σ01 = E(φ0φT ) describing covariances between unsampled

and sampled locations. For the non-stationary models, φ0 =∑K

k=1 a0kωk0, where a0k are

decreasing functions of the distance between new locations s0 and the centroid of the

subregion k.

Conditional on φ0i and β, Y0i are independent Bernoulli variates Y0i ∼ Ber(p0i) with ma-

laria prevalence at unsampled site s0i given by logit(p0i) = X t0iβ+φ0i. For the test locations

the predicted number of cases Yti arise from a Binomial distribution Yti ∼ Bin(Nti, pti),

where Nti is the number of tested children and pti is the predicted prevalence at test site

sti. The predictive distribution is numerically approximated by the average

1r

∑rq=1

[∏li=1 P (Y

(q)0i |β(q), φ

(q)0i

]P (φ

(q)0 |φ(q), σ2(q), ρ(q)),

where (β(q), φ(q), σ2(q), ρ(q)) are samples drawn from the posterior P (β, φ, σ2, ρ|Y , N ).

Chapter 4

Mapping malaria risk in West Africa

using a Bayesian nonparametric

non-stationary model

Gosoniu L.1 ,Vounatsou P.1, Sogoba N.2, Maire N.1, Smith T.1

1 Swiss Tropical Institute, Basel, Switzerland2 Malaria Research and Training Center, Universite du Mali, Bamako, Mali

This paper has been accepted for publication in Computational Statistics and Data Anal-

ysis, Special Issue on SPATIAL STATISTICS

56 Chapter 4. Mapping malaria risk in West Africa using a Bayesiannonparametric non-stationary model

Summary

Malaria transmission is highly influenced by environmental and climatic conditions but

their effects are often not linear. The climate-malaria relation is unlikely to be the same

over large areas covered by different agro-ecological zones. Similarly, spatial correlation in

malaria transmission arisen mainly due to spatially structured covariates (environmental

and human made factors), could vary across the agro-ecological zones, introducing non-

stationarity. Malaria prevalence data from West Africa extracted from the ”Mapping

Malaria Risk in Africa” database were analyzed to produce regional parasitaemia risk

maps. A non-stationary geostatistical model was developed assuming that the underlying

spatial process is a mixture of separate stationary processes within each zone. Non-linearity

in the environmental effects was modeled by separate P-splines in each agro-ecological zone.

The model allows smoothing at the borders between the zones. The P-splines approach

has better predictive ability than categorizing the covariates as an alternative of modeling

non-linearity. Model fit and prediction was handled within a Bayesian framework, using

Markov chain Monte Carlo (MCMC) simulations.

Keywords : Bayesian inference; geostatistics; malaria risk; Markov chain Monte Carlo; non-

stationarity; P-splines; kriging.

4.1 Introduction 57

4.1 Introduction

Plasmodium falciparum malaria is a major cause of mortality and morbidity in sub-Sahara

Africa. Malaria is a vector-borne disease and is transmitted from human to human by

female mosquitoes of the genus Anopheles. It is an environmental disease since its trans-

mission depends on the distribution and abundance of the mosquitoes, which are sensitive

to factors like temperature, rainfall and humidity. Climatic and environmental factors

play an important role in changes of the malaria distribution and endemicity, influencing

the survival and the development rate of both the parasite and the mosquito vector. The

relation between climatic/environmental factors and malaria transmission allows the pre-

diction of malaria risk at locations without observed data and therefore estimation of the

geographical distribution of the disease. Accurate maps of malaria transmission and ma-

laria risk are needed to estimate the burden of disease, to improve control and intervention

strategies and to optimize the use of limited resources in high-risk areas.

Reliable maps of malaria distribution are based on the availability of the disease data and on

appropriate methods for analyzes. The ”Mapping Malaria Risk in Africa” (MARA/ARMA,

1998) represents the most comprehensive database on malaria data in Africa and contains

malaria prevalence data collected over 10, 000 surveys from all available sources across

the whole continent. Using the MARA database a number of malaria risk maps have been

produced at both country and regional level. Initially, maps were developed using spatially

interpolated weather station data that were used to define climatic suitability for malaria

transmission (Craig et al., 1999). The mapping work continued with the application of

spatial statistical methods used to quantify the relations between environmental factors and

malaria risk and based on these relations to produce model-based predictions (Kleinschmidt

et al., 2000; Kleinschmidt et al., 2001; Gemperli et al., 2005; Gemperli et al., 2006).

Different assumptions and estimation approaches may result in different malaria risk maps.

Spatial models introduce at each data location an additional parameter on which the spatial

correlation is incorporated. Fitting these models is challenging because of the large number

of parameters. The first modeling efforts were based on maximum likelihood approaches

and could not estimate the environmental effects and the spatial correlation simultaneously

(Kleinschmidt et al., 2000). The Bayesian geostatistical models introduced by Diggle et

al. (1998) avoid the computational problems encountered in maximum likelihood-based

fitting. Most geostatistical modeling of malaria has been based on the assumption of

stationarity (Gemperli et al., 2005; Gemperli et al., 2006), which implies that the covariance


between any two points depends only on the distance between them. This assumption is not

justifiable when malariological indices are modeled, especially over large areas, since local

features related to human activities, land use, environment and vector ecology may affect

geographical correlation differently at various locations (Gemperli, 2003). Gosoniu et al.

(2006) analyzed malaria prevalence data in Mali under both assumptions of stationarity and

non-stationarity and performed model comparison which revealed that the non-stationary

model captured better the spatial correlation present in malaria data. Previous mapping

efforts over large areas assumed that the relation climate-malaria remains the same over the

entire area (Gemperli et al., 2006). However, the effect of environmental/climatic factors

on malaria risk depends on local ecological factors.

Modeling non-stationary spatial processes has recently received much attention (Banerjee

et al., 2004) due to computational advances in geostatistical model fit. Kernel convolu-

tion was introduced by Higdon et al. (1998) who convolve spatially evolving kernels and

Fuentes et al. (2002) who used a Gaussian process as a function of locations to obtain

a non-stationary covariance structure. Kim et al. (2005) used piecewise Gaussian pro-

cesses to model a non-stationary process by partitioning the study region in random tiles,

assuming an independent Gaussian stationary process within each tile and independence

between tiles. This approach has the computational advantage of inverting several covari-

ance matrices of smaller dimensions instead of the global large covariance matrix since the

covariance matrix is reduced to a block-diagonal form. Gemperli (2003) extended the work

of Kim et al. (2005) and model non-Gaussian malaria prevalence data using random tes-

sellations. This was the first time the non-stationarity issue was addressed in the mapping

malaria risk field. Although the assumption of independence between tiles facilitates the

matrix inversion, it ignores the spatial correlation between neighboring points located at

the edges and which belong to different tiles. To address the between-tile independence

problem Gosoniu et al. (2006) defined the spatial process as a mixture of tile-specific sta-

tionary spatial processes and demonstrated this methodology for a fixed space partitioning,

modeling malaria risk data in Mali. This approach is more appropriate when modeling

malaria data over large areas with fixed partitions defined by different ecological zones.

Another statistical issue which occurs in model-based malaria prevalence mapping is the

assumption of linearity between environmental factors and malaria risk which may not

always hold. The most popular method adopted to model the non-linearity is categoriz-

ing the covariates. The results are easy to understand, although it is unreasonable to

4.2 Data 59

conclude that the risk is increasing (or decreasing) abruptly as a category cut point is

crossed. There is considerable literature on nonparametric regression modeling, which al-

lows the shape of the relationship between outcome and covariates to be determined by

the data, whereas in the parametric framework the model determines this relationship.

Non-parametric modeling alternatives include kernel smoothing (Silverman, 1985; Har-

dle, 1990), local polynomial regression (Cleveland, 1979), fractional polynomials (Royston

and Altman, 1994) and spline smoothing. Spline approaches comprise regression splines

(Eubank, 1988), B-splines (de Boor, 1978) and penalized splines (P-splines). The latter

approach was first introduced by O’Sullivan (1986), but was popularized by Eilers and

Marx (1996). Bayesian approaches to P-splines allow simultaneous estimation of smooth

functions and smoothing parameters.

In this paper we develop non-stationary models to produce a smooth malaria risk map of

West Africa, modeling a non-linear relation between climate factors and malaria risk sepa-

rately in each agro-ecological zone. Non-linear environmental effects were modeled via cate-

gorizing the covariates and by P-splines. Modeling validation is applied to identify the best

approach to capture non-linearity. To model the non-stationary spatial process we extended

the work of Gosoniu et al. (2006) considering as fixed tiles the four agro-ecological zones in

West Africa. The computing time in the implementation of the geostatistical modeling was

reduced using a volunteer computer platform (www.malariacontrol.net) which is based on

Berkeley Open Infrastructure for Network Computing (http://boinc.berkeley.edu/). The

article is structured as follows. In Section 4.2 we describe the malaria data together with

the environmental and climatic data used in the analysis. The description of both the

nonparametric approach and Bayesian geostatistical non-stationary model as well as the

model validation approaches are provided in Section 4.3. The results are presented in

Section 4.4. Concluding remarks and suggestions for future work are given in Section 4.5.

4.2 Data

Malaria prevalence data were extracted from the MARA/ARMA database (MARA/ARMA,

1998). Only included surveys conducted after 1950 and on children between 1 and 10 years

old were included in the analysis. The final data set we analyzed was collected at 265

distinct locations over 374 surveys (Figure 4.1), including 56, 672 observed individuals.


Figure 4.1: Sampling locations of the MARA surveys included in the analyzes in WestAfrica with dot shading indicating the observed malaria prevalence. The diamonds indicatethe centroids of the four fixed tiles (AEZ) used to account for non-stationarity.

The environmental and climatic predictors used in this analysis are similar to the ones used

by Gemperli et al. (2006) for mapping malaria transmission in West Africa: minimum

and maximum temperature, amount of rainfall, the normalized difference vegetation index

(NDVI), the soil water storage (SWS) index, the length of the malaria transmission season,

the distance to the nearest water body and the land use.

Data on temperature and rainfall were extracted from the ”Topographic and climate

database for Africa” (Hutchinson et al., 1996). The data were obtained from weather sta-

tions between 1920 and 1980 and were extrapolated by fitting thin plate spline functions of

latitude, longitude and elevation to values of monthly mean daily minimum temperature,

daily maximum temperature and rainfall in the whole continent. Monthly averages over

the years with available data were calculated.

NDVI is used to indicate green vegetation cover and is calculated from the red and near

infra-red reflectance observed by the AVHRR (Advanced Very High Resolution Radiome-

ter) sensor on NOAA meteorological satellites (Agbu and James 1994) at a spatial resolu-

tion of 8km2. Index values can range from −1 to 1 with high values (0.3− 0.6) indicating

4.3 Spatial modeling 61

high levels of healthy (green) vegetation cover, whereas values near zero and negative val-

ues are associated with non-vegetated features such as barren surfaces and water. Monthly

NDVI values were obtained by averaging the maximum monthly index values over the

period 1985− 1995.

SWS estimates the amount of water that is stored in the soil within the plant’s root zone.

Monthly estimates of the SWS index were obtained at 5km2 resolution using the procedure

given by Droogers et al. (2001).

The length of malaria season was defined using the seasonality model of Gemperli et al.

(2006). They defined a region and month as suitable for malaria transmission when rainfall,

temperature and NDVI values are higher than pre-specified cut-offs .

Using Idrisi software (Clark Labs, Clark University) calculation for the distance to the

nearest water source was based on permanent rivers and lakes extracted from ”African

Data Sampler” (WRI, 1995).

Land use classification was based on land use/ land cover database maintained by the

United States Geological Survey and the NASA’s Distributed Active Archive Center. We

used the 24-category classification scheme described by Anderson et al. (1979) and re-

grouped them into six broad categories.

The region of West Africa was divided in four agro-ecological zones on the basis of the

period when the water is available for vegetative production on well-drained soils, according

to the procedure described in FAO (1978). The four agro-ecological zones are defined as

follows: Sahel (< 90 days), Sudan Savanna (90 − 165 days), Guinea Savanna (165 − 270

days) and Equatorial Forest (> 270 days).

4.3 Spatial modeling

The malaria prevalence data were treated as binomial data and modeled via the logistic

regression. Let Yi be the number of observed malaria cases out of Ni children tested at

location i, i = 1, . . . , n and Xi = (Xi1, Xi2, . . . , Xip)T be the vector of p associated envi-

ronmental predictors observed at location i. We assume that Yi are binomially distributed,

that is Yi ∼ Bin(Ni, pi) with parameter pi measuring malaria prevalence at location i.


4.3.1 Non-linearity of covariates effect

The relation between environmental factors and malaria risk may not be linear. It is

difficult to foresee the form of the relationship due to considerable residual variability in

the data. Parametric models are not appropriate, even if one assumes a non-linear relation

captured by a low-order polynomial, since the true relationship may not have a simple

polynomial form. In this analysis we focus on nonparametric regression models using P-

splines because they can be easily implemented in the Bayesian framework (Crainiceanu

et al., 2005).

The logit of pi is modeled non-parametrically logit(pi) =∑p

j=1 fj(Xij) + φi, where φi’s are

error terms, modeling between-location variation and f(·) is an unknown but a smooth

function of the environmental predictors. That is:

fj(Xij) =∑K

k=1 ukj|Xij − skj|3,

where uj = (u1j, . . . , uKj)T is the vector of regression coefficients, s1j < s2j < . . . <

sKj are fixed knots and |Xij − skj|3 is a truncated 3-rd order polynomial spline basis,

all corresponding to covariate Xj. In a simple regression spline approach the unknown

regression coefficients are estimated using standard maximum likelihood algorithms for

linear models. The problem in using this equation is the choice of the number and position

of the knots. Choosing a small number of knots would lead to a smooth function that is

not flexible enough to capture the variability in the data, while a large number of knots

may result in over-parametrization and over-fitting. To overcome these difficulties, one can

use a penalty on the spline coefficients u to achieve a smooth fit, that is we can penalize uj

by the quadratic form λuTj Duj, where λ is the smoothing parameter and D is a penalty

matrix chosen according to the data. The penalty controls the degree of smoothness. The

P-spline approach states that minimization of:∑ni=1 {logit(pi)−

∑pj=1 fj(xij)}2 + 1

λuT

j Duj

leads to a smooth vector of coefficients u. Eilers and Marx (1996) defined the penalty

function to be based on differences between neighboring spline coefficients.

The final model can be written as:

logit(p) = Zb + φ, with Cov

(b

φ

)=

(σ2

b IK 0

0 σ2φIn

),


where b = Q1/2u and Z = ZKQ−1/2 with ZK the matrix having the ith row ZKi =

{|xi−s1|3, . . . |xi−sK |3} and Q the matrix having the element (Q)ij = |si−sj|3 (Crainiceanu

et al., 2005).

In this paper we consider the knots to be based on sample quantiles of the covariates, but

one could take the knots to be equally spaced.

Since the environmental factors which influence malaria transmission are different in each

agro-ecological zone, a separate nonparametric model was derived for each of the four

zones.

The non-linear relation between malaria risk and environmental factors was also modeled

categorizing the risk factors. Scatter plots were used to choose the cutoffs points of the

categories.

4.3.2 Spatial correlation and non-stationarity

The parasitaemia risk at neighboring locations is influenced by similar environmental fac-

tors and therefore it is expected that the malaria risk varies similarly at locations within the

neighborhood. We model spatial heterogeneity via a geostatistical model by introducing a

random effect φi at each location i, that is logit(pi) = Zibi + φi. The spatial correlation is

modeled on these parameters as a function of the distance between locations. The design

matrix Z is the one obtained in the previous section.

Most geostatistical models assume stationarity of the spatial process, that is the spatial

correlation structure does not vary across the study area. This assumption is questionable,

especially when modeling malaria indices over large geographical areas. Differences in agro-

ecological zones, health systems and socio-economic indicators may change geographical

correlation differently at various locations.

Following the approach described in Gosoniu et al. (2006) the space was partitioned in

fixed tiles corresponding to the agro-ecological zones in West Africa. Thus the study area is

divided into 4 subregions and a stationary Gaussian process ωk is assumed in each subregion

k = 1, . . . , 4, that is ωk = (ωk1, . . . , ωkn)T ∼ N(0, Σk), with (Σk)ij = σ2kcorr(dij; ρk), where

corr is a parametric correlation function of the distance dij between locations i and j.

In most epidemiological applications, the exponential correlation function corr(dij; ρk) =

exp(−dij/ρk) is used, where ρk > 0 measures the rate of decrease of correlation with


distance and it is known as the range parameter of the spatial process. Parameter σ2k

measures within location variation and it is known as the sill of spatial process. Note that

the spatial parameters σ2k and ρk are specific for each agro-ecological zone k.

The spatial random effect φi at location i were modeled as a weighted sum of the tile-

specific stationary processes, that is φi =∑K

k=1 aikωki, where aik are decreasing functions

of the distance between location i and the centroid of the subregion k. Then φ is a non-

stationary spatial process φ ∼ N(0,∑K

k=1 AkΣkAk), where Ak is a diagonal matrix with

(Ak)ii = aik.

The Bayesian formulation of the model is completed by specifying prior distributions for

the model parameters β, σ2k and ρk (see Section 3.5).

The model described above have a large number of parameters. Bayesian computation

implemented via Markov chain Monte Carlo (MCMC) simulation methods enables simul-

taneously estimation of all model parameters together with their standard errors. More

details are given in Section 4.3.5.

4.3.3 Prediction

Bayesian kriging (Diggle et al., 1998) was employed to predict malaria risk at unsampled

locations. This approach has the advantage over the classical kriging that calculates the

predictive distribution of parasitaemia risk at new location and therefore, makes it possible

to estimate of prediction error.

Estimates of the malaria risk at any unsampled location s0 = (s01, s02, . . . , s0m)T were

obtained by the predictive distribution

P (Y0|Y , N ) =∫

P (Y0|β, φ0)P (ωk0|ωk, σ2k, ρk)P (β, ωk, σ

2k, ρk|Y,N) dβ dφ0 dωk, dσ2

k dρk,

where Y0 = (Y01, Y02, . . . , Y0m)T are the predicted number of children found with malaria

parasites in a blood sample at location s0, P (β, ωk, σ2k, ρk|Y , N ) is the posterior distribu-

tion and φ0 is the vector of random effects at new sites s0. Following the non-stationary

model, φ0 =∑4

k=1 a0kωk0, where a0k are decreasing functions of the distance between new

locations s0 and the centroid of the subregion k.

The distribution of ωk0 at unsampled locations given ωk at observed locations is normal


P (ωk0|ωk, σ2k, ρk) = N((Σ01)k(Σ11)

−1k ωk, (Σ00)k − (Σ01)k(Σ11)

−1k (Σ01)

Tk ),

where (Σ11)k is the covariance matrix created using the sampled locations s1, s2, . . . , sn,

(Σ00)k is the covariance matrix built by taking only the unsampled locations s01, s02, . . . , s0m

and (Σ01)k depicts the covariance between sampled and unsampled locations.

The predicted malaria prevalence at new location s0i is given by logit(p0i) = XT0iβ + φ0i,

where X0i are the environmental covariates corresponding to the unsampled location s0i.

4.3.4 Model validation

We considered two non-stationary Bayesian geostatistical models which address the non-

linearity of the covariates effect by using P-splines and categorized risk factors respectively.

The goodness-of-fit of each model was evaluated using the Deviance Information Criterion

(DIC) ( Spiegelhalter et al. 2002). To assess the best way of modeling non-linearity, model

fit was carried out on a randomly selected subset of 239 locations (training set). The

remaining 26 locations, comprising a simple random sample, were used for validation as

testing points (holdout set).

Following Gosoniu et al. (2006) the predictive ability of the two models was assessed by

using 1) a Bayesian ”p-value” analogue, 2) the probability coverage of the shortest credible

interval and 3) the Kullback-Leibler difference between observed and predicted prevalences.

For each test location we calculate the area of the predictive posterior distribution which is

more extreme than the observed data and we assert that the model which predicts better

is the one with the ”p-value” closer to 0.5.

Another approach to assess the accuracy of the prediction for the two models is to cal-

culate different coverages credible intervals of the posterior predictive distribution and to

compare the percentages of test locations with observed malaria prevalence falling in these

intervals. We calculated 12 credible intervals of the posterior predictive distribution at the

test locations with probability coverage equal to 5 %, 10 %, 20 %, 25 %, 30 %, 40 %, 50 %,

60 %,70 %, 80 %, 90 % and 95 %, respectively.

Finally, the Kullback-Leibler divergence from the observed prevalence to the predictive

posterior distribution was calculated as KL(j) =∑26

i=1 pobsi ∗ log(

pobsi

prep(j)i

), j = 1, . . . , 1000,

where pobsi is the observed prevalence at test site si and prep

i = (prep(1)i , . . . , p

rep(1000)i ) are

1000 replicated data from the predictive distribution at test location si.


The results of the model validation are presented in Section 4.4.

4.3.5 Implementation details

Although the choice of the number of knots is not essential for penalized splines (Ruppert,

2002), a minimum number of knots are needed to capture the spatial variability in the

data. We fixed the number of knots to 5 and we saw that increasing this number has no

significant change to the fit given by P-splines due to the penalty parameter. The P-spline

regression was performed in R version 2.4.0.

For the environmental variables with monthly values assigned we calculated summary

statistics over the months suitable for malaria transmission and assess which measure

leads to a better model fit in each agro-ecological zone. The calculated summary statistics

were maximum and average for the minimum temperature, maximum temperature, NDVI

and SWS and maximum, average and total for the rainfall. Table 4.1 depicts the measures

used for each environmental predictor and each agro-ecological zone.

Agro-ecological zone Environmental predictors

Min. Temp. Max. Temp. Rainfall NDVI SWS

Sahel Average Average Maximum Maximum Average

Sudan Savanna Average Average Average Maximum Average

Guinea Savanna Average Average Total Average Average

Equatorial Forest Average Average Total Maximum Maximum

Table 4.1: Measures of each environmental predictor which lead to a better model fit ineach agro-ecological zone.

The prior distributions used to complete the Bayesian formulation of the model were as

following. For the predictor’s coefficients βk we adopt non-informative Normal distribution

βk ∼ N(0, 102). We adopt an inverse Gamma prior for the variance parameters σ2k ∼

IG(a1, b1) and a Gamma prior distribution for the range parameters ρk ∼ G(a2, b2) with

the hyperparameters of these distributions chosen to have mean equal to 1 and variance

equal to 100. The model parameters were estimated by implementing the Gibbs sampler

(Gelfand and Smith, 1990) with five parallel chains, which requires simulating from the

conditional posterior distributions of all parameters, iteratively until convergence. The full

4.4 Results 67

conditional distributions of σ2k are inverse Gamma distributions and it is straightforward to

simulate from. The conditional posterior distributions of βk, ωk and ρk do not have known

forms. We simulate from these distributions using the Metropolis algorithm with a Normal

proposal distribution having the mean equal to the parameter estimate from the previous

Gibbs iteration and the variance equal to a fixed number, iteratively adapted to optimize

the acceptance rates. We have run a five chain sampler of 200, 000 iterations with a burn-in

of 10, 000 iterations and we assessed the convergence by examining the ergodic averages of

selected parameters. The analysis was implemented in Fortan 95 (Compaq Visual Fortran

Professional 6.6.0) using standard numerical libraries (NAG, The Numerical Algorithms

Group Ltd.).

4.4 Results

First, the results of the model validation are described because inference and mapping are

based on the model with the best predictive ability. Figure 4.2 shows the distribution of the

”p-values” of test locations and the distribution of the Kulback-Leibler difference measure

estimated by the P-spline model and the model with categorized covariates. The median

”p-value” of the former model is closer to 0.5, suggesting that this is the best model. The

P-spline model has also the smallest Kulback-Leibler value, supporting the results of the

previous validation approach.

Figure 4.2: The distribution of Bayesian p-values (left) and Kulback-Leibler differencemeasure (right) for the two non-stationary Bayesian geostatistical models. The box plotsdisplay the minimum, the 25th, 50th, 75th and the maximum of the distribution.


Figure 4.3 presents the percentages of test locations with malaria prevalence falling into

credible intervals of coverage ranging from 5% to 95% for both geostatistical models. For

the 95% credible interval the P-spline model included the highest precentage of test loca-

tions (96.15% versus 84.62%). Consistently, the P-sline model includes the highest per-

centage of observed locations in all coverage intervals.

Figure 4.3: Percentage of test locations with malaria prevalence falling in the 5%, 10%,20%, 25%, 30%, 40%, 50%, 60%, 70%, 80%, 90% and 95% credible intervals of the posteriorpredictive distribution.

Although the models differ in their predictive ability, the goodness of fit DIC measure does

not favor any of the models. In particular, the model with categorized covariates had a

DIC of 1636.1, while the P-spline model had a DIC of 1634.2.

Based on the results of the model validation, we use the P-spline non-stationary model for

estimating the relation climate-malaria and produce a smooth map of malaria risk.

Figures 4.4 through 4.7 show the non-linear effect of the environmental factors in all four

agro-ecological zones. The plots depict the posterior means and the 95% credible intervals.

The effect of each covariate changed from one agro-ecological zone to another, suggesting

that it was important to consider a separate nonparametric model in each zone.

4.4 Results 69

Figure 4.4 shows that in the Sahel zone the only covariates with significant effect on malaria

risk were distance to the nearest water body and season length since the credible intervals

of the posterior means do not include zero. The effect of distance to water on parasitaemia

risk is more or less constant. We detect a constant trend for the length of transmission

season up to 3 months and then a decreasing trend up to 5 months.

The impact of the environmental variables in the Sudan Savanna zone are presented in

Figure 4.5. The only credible interval that did not include zero was the one corresponding

to the posterior mean of the minimum temperature, therefore we conclude that this was

the only variable with a significant effect on the malaria risk in Sudan Savanna area. We

notice that minimum temperature had a constant effect between 19◦C and 21◦C and then

it showed an increasing risk effect from 21◦C to 25◦C.

In Guinea Savanna the environmental factors significant associated with the parasitaemia

risk were: distance to the nearest water body, maximum temperature, rainfall and length

of the malaria transmission season (Figure 4.6). The distance to the nearest water body

had a slightly increasing effect up to 10 km, indicating high malaria prevalence in areas

within 10 km away from a water source, which in this study was considered permanent

river or lake. The risk of malaria decreased in regions 10 km further away from a water

body. The maximum temperature had a decreasing effect in Guinea Savanna between

28◦C and 32◦C. As expected, the effect of rainfall on malaria prevalence was not linear.

We observe an increase in parasitaemia prevalence when rainfall was between 800 mm and

1300 mm. The risk of malaria decreased when the amount of rainfall exceeded 1300 mm

since excessive rainfall may flush out the eggs or larvae out of the pools impeding the

development of mosquito eggs or larvae. Malaria prevalence slightly decreased for length

of malaria transmission season between 6 and 8 months and increased when the malaria

transmission season exceeds 9 months.

Figure 4.7 shows the non-linear effect of the predictors on malaria prevalence in Equatorial

Forest. The only variable with a significant risk effect was rainfall, which showed an

increased effect on malaria risk. All the other environmental factors had zero included in

the credible intervals of the posterior means, thus their influence is minimal.

We notice that for some variables the credible intervals tend to widen at the tails of the

splines. This could be explained by the fact that only few data locations had extreme

values for those variables.


Figure 4.4: Estimated non-linear effect (P-spline) of environmental factors on malaria riskin West Africa, Sahel. The posterior mean (pink) and the 95% credible interval are shown.

4.4 Results 71

Figure 4.5: Estimated non-linear effect (P-spline) of environmental factors on malaria riskin West Africa, Sudan Savanna. The posterior mean (pink) and the 95% credible intervalare shown.


Figure 4.6: Estimated non-linear effect (P-spline) of environmental factors on malaria riskin West Africa, Guinea Savanna. The posterior mean (pink) and the 95% credible intervalare shown.

4.4 Results 73

Figure 4.7: Estimated non-linear effect (P-spline) of environmental factors on malaria riskin West Africa, Equatorial Forest. The posterior mean (pink) and the 95% credible intervalare shown.


In Table 4.2 we present the posterior estimates (odds ratio and credible intervals) cor-

responding to the land use regression coefficients in each agro-ecological zone obtained

from the Bayesian geostatistical non-stationary model. In Sahel and Guinea Savanna the

land use was not significantly associated with the malaria risk when spatial correlation was

taken into account. The odds of malaria for locations situated around cropland were signifi-

cantly higher than those in urban area in Sudan Savanna (OR=4.13, 95% credible interval:

1.25, 19.79). In Equatorial Forest malaria odds were significantly higher in Grass/Shrub

land/Savanna (OR=4.06, 95% credible interval: 1.12, 19.63) and Forest (OR=3.53, 95%

credible interval: 1.03, 13.78) compared with the urban areas.

Agro-ecological zone Land use OR 95% CIa

Sahel Cropland 1.00

Grass/Shrub land/Savanna 0.37 (0.02, 3.13)

Water bodies 0.06 (0.00, 1.71)

Urban area 0.17 (0.02, 2.16)

Sudan Savanna Urban area 1.00

Cropland 4.13 (1.25, 19.79)


Water bodies 2.05 (0.26, 14.22)

Wetland 1.86 (0.24, 19.77)

Guinea Savanna Urban area 1.00


Forest 0.22 (0.01, 2.24)

Water bodies 0.61 (0.00, 15.12)

Equatorial Forest Urban area 1.00

Cropland 2.25 (0.13, 78.49)


Forest 3.53 (1.03, 13.78)

Water bodies 2.48 (0.36, 17.01)

Wetland 1.13 (0.14, 8.87)

a : Credible intervals (or posterior intervals).

Table 4.2: Posterior estimates for land use coefficients. The land use is the only categoricalvariable in the P-spline non-stationary model.

4.4 Results 75

Posterior estimates of the spatial parameters (spatial variance and decay parameter) in the

four zones are shown in Table 4.3. In Sahel the posterior median of ρ was equal to 6.23

(95% credible interval: 1.15, 22.31), which in the current exponential setting corresponds

to a minimum distance for which the spatial correlation becomes negligible of 0.48 km

(95% credible interval: 0.13, 2.61). In the other 3 zones the decay parameters were similar

and indicated ranges of 4.23 km (95% credible interval: 0.61, 28.64) in Sudan Savanna,

4.58 km (95% credible interval: 0.56, 98.84) in Guinea Savanna and 4.86 km (95% credible

interval: 0.54, 44.78) in Equatorial Forest. We conclude that the spatial correlation is weak

in the Sahel zone and strong in the other agro-ecological zones. The spatial variance varied

from 0.83 (95% credible interval: 0.34, 1.67) in the Sahel to 3.33 (95% credible interval:

1.48, 7.64) in Guinea Savanna.

Agro-ecological zone Spatial parameter Median 95% CIa

Sahel σ2 0.83 (0.34, 1.67)

ρb 6.23 (1.15, 22.31)

Sudan Savanna σ2 1.31 (0.55, 2.98)

ρb 0.70 (0.10, 4.88)

Guinea Savanna σ2 3.33 (1.48, 7.64)

ρb 0.66 (0.03, 5.39)

Equatorial Forest σ2 1.78 (1.10, 3.02)

ρb 0.62 (0.07, 5.53)a : Credible intervals (or posterior intervals).b : Based on ρ we calculate the range parameter 3/ρ (in km).

Table 4.3: Posterior estimates of spatial parameters.

The map of predicted malaria prevalence for sub-Sahara West Africa is shown in Figure 4.8.

We find a relatively high malaria risk with only few exceptions. High levels of prevalence

were predicted in the center-east of Senegal, the north of Ghana, the south of Togo, some

areas in the west and east of Nigeria and north-west of the Democratic Republic of Congo.

In addition, the areas along the Atlantic Ocean were estimated to have high malaria risk.

Low levels of malaria prevalence were observed in the north and west of Senegal, Guinea

Bissau, Guinea, north-west of Cote d’Ivoire, the south of Ghana, the north and north-east

of Nigeria, the north of Cameroon, a small region in center Gabon and Republic of Congo.

The prediction error from the Bayesian geostatistical non-stationary model is depicted in

Figure 4.9.


Figure 4.8: Map of predicted malaria prevalence in children 1− 10 years for West Africa.

4.5 Discussion

Malaria is an environmental disease and its endemicity depends on the density and infec-

tivity of anopheline vectors. These are highly influenced by meteorological variables, but

the relation is often not linear. In this analysis we modeled this relation using Bayesian

P-splines. The comparison between Bayesian P-splines and the widely used method which

considers the covariates as categorical variables (Section 4.3.4) shows that the former model

fits better the relation between malaria risk and environmental factors. It is the first time

the non-parametric spline approach is used to obtain a smooth map of malaria prevalence.

Previous mapping efforts in West Africa assumed the same climate-malaria relation across

the agro-ecological zones and modeled the non-linearity by using polynomial functions

(Kleinschmidt et al., 2001) and by comparing different functional forms of the predictors

(Gemperli et al., 2006). However, the polynomial functions and the pre-specified functional

forms may not be able to capture the true underlying relations.

4.5 Discussion 77

Figure 4.9: Map of prediction error.

We assume that the environmental factors have a different influence on malaria transmis-

sion in the four agro-ecological zones and we consider a separate nonparametric model in

each zone. Kleinschmidt et al. (2001) addressed this issue by a non-Bayesian spatial model.

However, the resulting map showed discontinuities at the borders between the zones, which

were further smoothed. This problem was prevented in our case since we used a mixture

of spatial processes over the whole area.

In the Sahel, the temperature is high and the amount of rainfall is very low. Thus mosquito

populations increase rapidly at the onset of the rain, because of short vector developmental

cycles (Picq et al., 1992). Consequently, the intensity of transmission may depend on the

number of rainy months. This supports the significant association between the length of

transmission season and malaria prevalence found in the Sahel.

The temperature affects the transmission cycle in many different ways, but the effects on


the duration of the sporogonic cycle of the parasite and vector survival are particularly

important. In Sudan Savanna the minimum temperature had a significant effect on the

parasitaemia risk, with values above 21◦C indicating an increased risk of malaria. Our

findings correspond to the ones of Craig et al. (1999) who showed that temperatures above

22◦C are suitable for stable malaria transmission.

The Guinea Savanna represents the optimum climate envelope for malaria transmission

because it offers favorable conditions for An. gambiae s.l. and An. funestus mosquitoes.

Usually, the limiting factor of transmission in this zone is the heavy rainfall which flushes

out many larvae and pupae out of the pools or decreases the temperatures. This was also

reflected by our results.

Malaria data observed over large areas have non-stationary characteristics. Ignoring this

feature could lead to a misspecification of the spatial correlation and therefore to wrong

estimates of the standard error of both the covariates and the prediction. In our recent

work (Gosoniu et al., 2006) we addressed non-stationarity in mapping malaria risk in Mali

by dividing the study area into fixed number of tiles, assuming a separate stationary spatial

process in each tile and correlation between tiles. In this paper the fixed tiles correspond

to the four different agro-ecological zones in West Africa. This method is appropriate for

malaria mapping over large areas with a clear way of defining the partitioning. We are

currently extending this model by assuming random space partitioning to accommodate

mapping over areas with no clear way of finding a fixed tessellation.

Comparing our malaria risk map for children 1− 10 years old in West Africa with those of

Kleinschmidt et al. (2001) and Gemperli et al. (2006) we observe similar pattern of malaria

prevalence. We all estimated high prevalence at the border between Mali and Senegal, the

north part of Ghana, the south area of Togo and the west part of Cameroon. Low levels of

prevalence were estimated by all three maps in the north-west of Senegal, Guinea Bissau

and the north of Cameroon. We notice several differences in the three maps: in Guinea

we estimated a low prevalence (< 0.2), Gemperli et al. (2006) estimated a prevalence

between 0.3 and 0.4, while Kleinschmidt et al. (2001) showed a prevalence between 0.3

and 0.7. In Sierra Leone both our map and the one of Kleinschmidt et al. (2001) show

a high prevalence, whereas Gemperli et al. (2006) predicted a lower level of malaria risk.

Our map and Kleinschmidt et al. (2001) map estimate a low prevalence in south of Ghana,

while the map of Gemperli et al. (2006) shows higher risk.

4.5 Discussion 79

Gemperli et al. (2006) used the malaria transmission Garki model (Molineaux and Gram-

iccia, 1980) to produce age and seasonality adjusted maps from the MARA survey data

which are heterogeneous in age and seasonality across locations. In this study we discarded

the surveys carried out on population outside the range of 1 to 10 years old. Currently we

are employing the method developed in this paper and newly developed stochastic trans-

mission model (Smith et al., 2006) to produce age-adjusted maps in West and Central

Africa, making use of all the available MARA data.

Although we used a mixture of spatial processes over the area of interest, the map of

predicted malaria risk show discontinuities between the agro-ecological zones on the right

hand side. This limitation could be explained by the position of the fixed centers of the

tiles. We have chosen the coordinates of the fixed centers as the average of the coordinates

of the data points belonging to the specific tile. However, the distribution of the survey

locations was higher in the east side of the study area, therefore the centers of the tiles were

shifted towards that part of the region. The weights used in the calculation of location-

specific random effects were based on the distance between the fixed centers of the tiles

and the specific points, hence for the points in the west side of the map the weights were

very small and unable to smooth the tile-specific spatial processes at the borders. In this

situation (and usually in the case when the tiles have more a rectangular rather than

square or circle shape) one may consider more than one fixed points as ”centers” per tile

and calculate the weight for each location from the nearest ”center” of the tile. We are

currently exploring this approach by re-analyzing malaria data in West and Central Africa.

Acknowledgments

The authors would like to thank the MARA/ARMA collaboration for making the malaria

data available. We are grateful to the volunteer computer platform malariacontrol.net and

to all the people around the world who made available their computers to help us predict

malaria distribution in a feasible period of time. This work was supported by the Swiss

National Foundation grant Nr.3252B0-102136/1.

Chapter 5

Non-stationary partition modeling of

geostatistical data for malaria risk

mapping

Gosoniu L.1 ,Vounatsou P.1

1 Swiss Tropical Institute, Basel, Switzerland

This paper has been submitted to Journal of Applied Statistics.

82 Chapter 5. Non-stationary partition modeling of geostatistical datafor malaria risk mapping

Summary

The most common assumption in geostatistical modeling of malaria is stationarity, that is

spatial correlation is a function of the distance between locations and independent of the

locations themselves. However, local factors (environmental or human related activities)

may influence geographical dependence in malaria transmission differently at different loca-

tions, introducing non-stationarity. Ignoring this characteristic in malaria spatial modeling

may lead to inaccurate estimates of the standard errors for both the covariate effects and

the predictions. In this paper a model based on random Voronoi tessellation that takes

into account non-stationarity was developed. In particular, the spatial domain was parti-

tioned into sub-regions (tiles), a stationary spatial process was assumed within each tile

and between-tile correlation was taken into account. The number and configuration of the

sub-regions are treated as random parameters in the model and inference is made using

reversible jump Markov chain Monte Carlo (RJMCMC) simulation. This methodology was

applied to analyze malaria survey data from Mali and to produce a country-level smooth

map of malaria risk.

Keywords : Bayesian inference; geostatistics; kriging; MARA; malaria risk; prevalence data;

non-stationarity; reversible jump Markov chain Monte Carlo; Voronoi tessellation.

5.1 Introduction 83

5.1 Introduction

Malaria is one of the most common infectious diseases and a major international public

health problem. According to the World Health Organization, every year between 300

and 500 million people are infected with malaria. Most cases occur in sub-Sahara Africa

(with approximately 2 million deaths each year), but many infections persist there in an

asymptomatic state. Accurate estimates of the burden of malaria are needed for evidence-

based planning of malaria control. In endemic areas malaria burden is best measured by

the age-specific prevalence of infection, but for most African countries no comprehensive

surveys have been carried out and it is needed to use ad hoc collections of local surveys.

The survey data are correlated in space, therefore geostatistical methods are the most

appropriate for obtaining smooth maps of malaria risk and estimates of number of people

at risk of malaria. Malaria is an environmental disease because its transmission depends

on the distribution and abundance of mosquitoes, which are sensitive to environmental and

climatic factors. Estimating the environment-parasitaemia relation we can predict malaria

transmission at locations where data are not available .

In geostatistics it is commonly assumed that the spatial dependence between two points

is a function of only the separation distance and independent of the absolute locations.

However, malaria data observed over large areas have non-stationary characteristics be-

cause spatial correlation may be influenced differently at various locations due to local

characteristics like environmental factors, intervention measures, mosquito ecology, human

activities (e.g. irrigation, dam construction), health services etc. Ignoring this feature

could lead to wrong estimates of the standard error for both covariates coefficients and

predictions (Gosoniu et al., 2006).

In recent years, much attention has been concentrated toward the development of models

that allow for non-stationary spatial covariance structure. Obled and Creutin (1986) pro-

posed the empirical orthogonal function (EOF) approach based on an eigenfunction expan-

sion of the covariance function and is very popular in the spatial analysis of environmental

data. Nychka et al. (2002) replaced the basis of the orthogonal functions with wavelet

basis functions. Although these methods are very flexible, they are not easily interpreted

from a geostatistical point of view. The spatial deformation method introduced by Samp-

son and Guttorp (1992) assumes that the spatial process is stationary and isotropic only

after some nonlinear transformation of the original sampling space. A Bayesian framework

for the deformation method was proposed independently, by Damian et al. (2001) and


Schmidt and O’Hagan (2003). Haas (1990, 1995) modeled non-stationary spatial processes

using a moving window approach which assumes stationarity only within the window con-

structed around the locations where kriging is performed. Higdon et al. (1998) assumed

that any stationary Gaussian process can be expressed as the convolution of a Gaussian

white noise process with a kernel function (i.e. bivariate Gaussian density function). They

account for non-stationarity by allowing the kernel to vary smoothly over space. Kim et al.

(2005) modeled non-stationary Gaussian data by partitioning the region of interest into

sub-regions via a Voronoi tessellation, assuming stationarity within each tile but across

sub-regions the data are assumed to be independent. The shape and the number of sub-

regions are estimated by the data. Fuentes (2001) represents a non-stationary process as

a weighted average of locally stationary processes defined within disjoint sub-regions of

the area of interest. The weights of the local processes are based on kernel functions. The

shape of the sub-regions is known a priori and the number of sub-regions is chosen using an

AIC or a BIC criterion. Banerjee et al. (2004) replaced the kernels by decreasing functions

of the distance between the data points and the centroids of the sub-regions.

In the field of disease mapping the non-stationarity issue was previously addressed by

Gosoniu et al. (2006) who extended the work of Banerjee et al. (2004) to model non-

Gaussian malaria prevalence data in Mali. Raso et al. (2005) applied the model developed

by Gosoniu et al. (2006) to map hookworm infection prevalence in western Cote d’Ivoire.

In this paper we further extend our previous work (Gosoniu et al., 2006) by considering

random rather than fixed tiles to map malaria prevalence in Mali. In particular, we parti-

tion the spatial domain into sub-regions using a Voronoi tiling, assume stationary spatial

covariance structure within sub-regions and correlation between them. The shape and the

number of tiles are treated as unknown variables and inference is made using reversible

jump Markov chain Monte Carlo (RJMCMC) (Green, 1995). A description of the malaria

data which motivated this work and the environmental variables used as predictors is given

in section 5.2. Section 5.3 describes the Bayesian partition model and details of the im-

plementation by RJMCMC are given in Section 5.4. We analyzed the malaria prevalence

data in Section 5.5. The paper ends with final remarks and conclusions in Section 5.6.

5.2 Motivating example: mapping malaria risk in Mali 85

5.2 Motivating example: mapping malaria risk in Mali

The data which motivated this work are malaria prevalence data extracted from the ”Map-

ping Malaria Risk in Africa” (MARA/ARMA,1998) database. MARA is the most com-

prehensive database of malaria prevalence data in Africa containing over 10.000 distinct

age-specific prevalence values extracted from published and unpublished sources across the

whole continent. We selected prevalence data from malaria surveys carried out between

1977 and 1995 at 89 locations in Mali on children between 1 and 10 years old (Figure

5.1). The children were considered malaria positive if Plasmodium falciparum was found

in the blood smears collected. The size of the surveys varied from 14 to 4835. There

were 43.492 sampled children and 44% were found malaria positives. Environmental data

used as malaria predictors such as rainfall, maximum and minimum temperature, vegeta-

tion index, were extracted from remote sensing. Details on sources and resolutions of the

environmental data are given in Gosoniu et al. (2006).

Figure 5.1: Sampling locations in sub-Saharan Mali with dot shading indicating the ob-served malaria prevalence.


5.3 Modeling non-stationarity via dependent spatial

processes

At each location si ∈ A ⊂ R2, i = 1, . . . , n the data are available in the form of pairs, giving

the number of children tested Ni and the number of those found positive to P.falciparum

parasitaemia Yi. These are typical binomial data and modeled via logistic regression. Let

Xi = (Xi1, . . . , Xip)T be a collection of p environmental explanatory variables available

at each location si. We model the relation between the environmental covariates Xi and

the malaria risk pi at location si via the logistic regression, that is logit(pi) = XTi β,

where β = (β1, β2, . . . , βp)T is the vector of regression coefficients. Parasitaemia risk at

close geographical proximity is influenced by similar factors therefore we can not assume

independence of observations. To account for the spatial variation present in the data

we introduce at each location si a random effect φi, that is logit(pi) = XTi β + φi and

model the spatial correlation on these parameters. Most geostatistical models assume

that φ = (φ1, φ2, . . . , φn)T models a latent stationary spatial process with spatial correla-

tion structure that does not vary across the study area. This assumption is questionable,

especially when modeling malaria indices over large geographical areas. Differences in agro-

ecological zones, health systems and socio-economic indicators may change geographical

correlation differently at various locations, so it cannot be expressed as a simple homoge-

neous function of the distance between the locations.

We relax the assumption of stationarity by defining a spatial partitioning via Voronoi

tessellation. In particular, we divide the area of interest A into several subregions (tiles)

Tk, k = 1, . . . , K, such that A =⋃K

k=1 Tk and Ti ∩ Tj = �,∀i 6= j. Denoting by c =

(c1, . . . , cK)T the centroids of the Voronoi tiles T = (T1, . . . , TK)T , we define the tile Tk as

Tk = {si ∈ A|d(si, ck) < d(si, cl),∀l 6= k}, where d is a distance measure (usually Euclidean

distance). The number and centroids of the tiles are treated as unknown parameters of the

model.

Spatial partitioning via Voronoi tiles was proposed by Kim et al. (2005) to model non-

stationary spatial processes. However, the authors assumed a separate stationary process

in each tile and independence of the data across the tiles.

Although the assumption of independence between tiles facilitates the matrix inversion by

converting the spatial covariance matrix into block diagonal form, it ignores the spatial

correlation between neighboring points located in different tiles. In this paper we extend

5.4 Implementation details 87

the random tessellation model to address the between-tile correlation by defining the spatial

process as a mixture of tile-specific stationary spatial processes.

In each tile k we assume a stationary spatial process ωk, that is ωk = (ωk1, . . . , ωkn)T ∼N(0, Σk). Here (Σk)ij = σ2

kcorr(dij; ρk), where σ2k corresponds to the sill of the spatial

process and corr is a parametric correlation function of the distance dij between locations

si and sj belonging to the area A. For the current example we choose the exponential

correlation function corr(dij; ρk) = exp(−dijρk), where ρk captures the rate of correlation

decay with distance and is known as the range parameter. We then model the spatial

random effect φi at location si as a weighted sum of the tile-specific stationary processes,

that is φi =∑K

k=1 aikωki, where aik are decreasing functions of the distance between location

si and the centroid of the subregion k. Then φ is a non-stationary spatial process φ ∼N(0,

∑Kk=1 AkΣkAk), where Ak is a diagonal matrix with (Ak)ii = aik. The number K and

the centroids c = (c1, . . . , cK)T of the tiles are unknown parameters of the geostatistical

model.

5.4 Implementation details

5.4.1 Model fit

We assume that the prior distribution for the number of tiles K has the form

Pr(K) ∝ (1 − α)K , with the parameter α ∈ [0, 1) pre-defined. Knorr-Held and Rasser

(2000) recommend small values for α to specify a non-informative prior. Given a number

of tiles k, for the vector of tile centers ck = (c1, . . . , ck)T we assume the prior distribution

Pr(ck/k) = (n−k)!n!

.

For the regression coefficients β we adopt a non-informative uniform prior distribution

Pr(β) = U(−∞,∞). For the spatial parameters σ2k and ρk we choose inverse gamma and

gamma prior distributions respectively, that is Pr(σ2k) = IG(a1, b1) and Pr(ρk) = G(a2, b2).

The hyperparameters a1, b1 and a2, b2 are chosen so that the prior distributions are proper

but non-informative so that the inference is driven by the data.

The unknown parameters are estimated using the Metropolis-Hastings algorithm (Hastings,

1970). These methods are useful for sampling from the posterior distribution when the

dimension of the parameter vector is fixed. In our application the number of tiles is not

fixed, therefore at each iteration the number of parameters is unknown. To estimate the


parameters of our model, reversible jump MCMC (RJMCMC) was used. The RJMCMC

algorithm introduced by Green (1995) is an extension of the Metropolis-Hastings algorithm

which allows simulation from posterior distribution on spaces of varying dimensions. The

method is called reversible jump due to the ability of the Markov Chain to jump between

parameter spaces of different dimensions. At each iteration t we consider four possible

steps, that is: STAY, BIRTH, DEATH and MOVE. Each of these steps are randomly

chosen with probabilities Qs, Qb, Qd, Qm, such that Qs + Qb + Qd + Qm = 1.

In the STAY step the number of tiles k and the centroids ck remain at the current value and

we estimate the remaining model parameters using the Gibbs sampling. The only conjugate

distributions are the full conditional distributions of σ2k which are inverse gamma and

simulation from them is straightforward. The remaining parameters can not be sampled

directly from the full conditionals, hence we employ a random walk Metropolis algorithm

(Tierney, 1994) having a Normal proposal density with mean equal to the estimate of the

corresponding parameter from the previous Gibbs iteration and variance equal to a fixed

number, iteratively adapted during the burn-in period to optimize the acceptance rates.

In the BIRTH step the number of tiles increases by one, adding a new center cK+1, uniformly

selected form the n−K points which are not already centers. In this case the dimension

of the vector parameter changes from p + n + 2K to p + n + 2K + 2 + n by introduc-

ing 2 additional spatial parameters σ2K+1 and ρK+1 and a new stationary spatial process

ωK+1 = (ω(K+1)1, . . . , ω(K+1)n)T ∼ N(0, ΣK+1), where (ΣK+1)ij = σ2K+1exp(−dijρK+1),

corresponding to the new tile. To match the dimensions of the parameter spaces between

successive iterations with variable number of parameters we introduce the parameters

u(t−1) = (u(t−1)

σ2 , u(t−1)ρ , u

(t−1)ω )T . u

(t−1)

σ2 and u(t−1)ρ are generated from a gamma distribu-

tion and u(t−1)ω from a multivariate Normal distribution, independently from the value of

parameters at previous iteration. We then define σ2K+1 = f(u

(t−1)

σ2 ), ρK+1 = f(u(t−1)ρ ) and

ωK+1 = f(u(t−1)ω ), with f the identity function 1. Then the location-specific random effects

at a birth step will be φnewi =

∑K+1k=1 aikωki, i = 1, . . . , n. The birth step is accepted with

probability αb = min(1, LbAbPb |Jb|), where

Lb = L(Y ,N ,β,φnew)L(Y ,N ,β,φ)

is the likelihood ratio,

Ab =Pr(ω

(t+1)K+1 )Pr(σ

2(t+1)K+1 )Pr(ρ

(t+1)K+1 )

1Pr(K+1)

Pr(K)1

(n−K)

is the prior ratio and

5.4 Implementation details 89

Pb = Qd

Qb

(n−K)Γ(aσ2 ,bσ2 )Γ(aρ,bρ)N(0,ΣK+1)

.

is the proposal ratio. aσ2 , bσ2 and aρ, bρ are the parameters of the proposal gamma distri-

bution for the spatial parameters σ2 and ρ, respectively. The determinant of the Jacobian

resulting from the potential change of dimension of the parameter vector |Jb| = 1 since we

draw the new spatial parameters independent of the current parameters.

In the DEATH step the number of tiles decreases by deleting a center, uniformly chosen

from the K existing ones. The spatial parameters corresponding to the deleted tile are σ2d

and ρd. We also delete the stationary spatial process ωd = (ω(d1, . . . , ωdn)T ∼ N(0, Σd),

where (Σd)ij = σ2dexp(−dijρd). Similarly to the birth step, the death step is accepted with

probability αd = min(1, LdAdPd |Jd|). The likelihood ratio is

Ld = L(Y ,N ,β,φnew)L(Y ,N ,β,φ)

where φnewi =

∑K−1k=1 aikωki, i = 1, . . . , n. The determinant of the Jacobian |Jd| is equal to

1. The prior ratio and the proposal ratio have the same form as the corresponding birth

step, except that the ratio is inverted, that is:

Ad = 1

Pr(ω(t)d )Pr(σ

2(t)d )Pr(ρ

(t)d )

Pr(K−1)Pr(K)

(n−K+1)1

and

Pd = Qb

Qd

Γ(aσ2 ,bσ2 )Γ(aρ,bρ)N(0,Σd)

(n−K+1)

In the MOVE step the number of tiles K remains at the current value and we uniformly

select a center from the current ones and propose a new location for it by uniformly

sampling a data point from the n − K available points. The tessellation structure is

modified but the values of the parameters are not altered. The MOVE step is accepted with

probability αm = min(1, LmAmPm |Jm|), with proposal ratio Pm = 1 and the determinant

of the Jacobian |Jm| = 1. The likelihood ratio is defined by:

Ld = L(Y ,N ,β,φnew)L(Y ,N ,β,φ)

with φnewi =

∑Kk=1 aikωki, i = 1, . . . , n. In this case ak and ωk are the weights and the tile-

specific random effects corresponding to the new tessellation. The prior ratio is Am = 1.


5.4.2 Prediction

We predict malaria risk at a set of unsampled locations s(0) = (s(0)1 , s

(0)2 , . . . , s

(0)l )T by

using Bayesian kriging. In particular, we obtain estimates of the number of cases Y (0) =

(Y(0)1 , Y

(0)2 , . . . , Y

(0)l )T at locations s(0) from the predictive distribution

P (Y (0)|Y , N ) =∫∫∫P (Y (0)|β, ω

(0)k )P (ω

(0)k |ωk, σ

2, ρ, K, ck)P (β, ωk, σ2, ρ, K, ck|Y , N ) dβ dω

(0)k dωk dσ2 dρ dK dck,

where P (β, ωk, σ2, ρ, K, ck|Y , N ) is the posterior distribution and ω

(0)k = (ω

(0)k1 , . . . , ω

(0)kl )T

is the vector of tile-specific random effects at new site s(0). The distribution of ω(0)ki given

ωk = (ωk1, . . . , ωkn)T at observed locations is Normal

P (ω(0)ki |ωk, σ

2, ρ, K, ck) = N(Σ01k (Σ11

k )−1ωk, Σ00k − Σ01

k (Σ11k )−1(Σ01

k )T ),

with Σ11k = E(ωkω

Tk ) the covariance matrix built by including only the sampled locations

s1, s2, . . . , sn, Σ00k = E(ω

(0)k ω

(0)Tk ) the covariance matrix formed by taking only the new

locations s(0)1 , s

(0)2 , . . . , s

(0)l and Σ01

k = E(ω(0)k ωT

k ) describing covariances between unsampled

and sampled locations. The location-specific random effect at new site s(0)i is calculated

as φ(0)i =

∑Kk=1 a

(0)ik ω

(0)ki , where a

(0)ik are decreasing functions of the distance between new

location s(0)i and the centroid of tile k. Conditional on φ

(0)i and β, Y

(0)i are independent

Bernoulli variates p(Y(0)i |β, φ

(0)i ) ∼ Ber(p

(0)i ) with logit(p

(0)i ) = X

(0)Ti β + φ

(0)i , where X(0)

are environmental covariates at new locations s(0).

5.5 Analysis of the malaria prevalence data

The performance of the spatial logistic model described in Section 5.3 is illustrated on

malaria prevalence data presented in Section 5.2. The relation between the malaria risk and

the environmental factors is not linear (Chapter 4). A preliminary non-spatial analysis was

run to find the best combination and transformation of the environmental factors based on

the AIC. The factors and their transformations included in the analysis were: Normalized

Difference Vegetation Index (NDVI), NDVI squared, length of malaria season, amount of

rainfall, maximum temperature, squared maximum temperature, minimum temperature,

squared minimum temperature, distance to the nearest water body and squared distance

to the nearest water body.

We used Metropolis-Hastings algorithm to sample from the full conditional distributions for

5.5 Analysis of the malaria prevalence data 91

all model parameters. The starting values for the regression coefficients β were set equal to

the estimates of the non-spatial logistic regression. The location-specific random effects φi,

i = 1, . . . , n were all initialized with 0. The initial values for the spatial variances σ2k were

fixed to 0.1 and for the decay parameters ρk to 1.0, k = 1, . . . , K. The probabilities Qs,

Qb, Qd, Qm of the RJMCMC were set to 0.4, 0.2, 0.2 and 0.2 respectively. The parameter

α of the prior distribution of K was fixed to 0.1 to obtain a non-informative prior.

We ran a single chain of 120, 000 iterations with a burn-in of 5, 000 iterations. Conver-

gence was assessed by inspection of ergodic averages of selected model parameters. After

convergence we collected a sample of size 1, 000 from the posterior distribution by taking

every 10th value from the chain to avoid autocorrelation in the sample.

Table 5.1 shows the posterior estimates of the effects of environmental predictors. The

environmental covariates significantly related to malaria risk were rainfall, maximum tem-

perature and distance from the water. We found a negative association between malaria

prevalence and the average amount of rainfall during the malaria transmission season. A

similar negative association was observed between the average of maximum monthly tem-

perature during the transmission season and the malaria risk. The coefficient corresponding

to the distance from nearest water body indicates high malaria risk in areas away from

water sources. A similar result was found by Gemperli (2003) in the analysis of MARA

data from Mali.

The posterior frequency for the number of partitions is summarized in Figure 5.2. The

most frequent tessellation favors two tiles, suggesting two spatial processes. Figure 5.3

displays the lower and upper quartiles and the median of the posterior distribution for the

spatial covariance parameters σ2 and ρ. The parameter ρ varies from 0.0154 to 0.2781,

which in our exponential setting indicates that the minimum distance for which the spatial

correlation is less than 5% varies between 10.79 and 194.81 kilometers.


Variable Median 95% CIa

Intercept 0.65 (0.09, 1.3)

Log(NDVI) 0.63 (-0.47, 1.04)

Log(NDVI)2 -0.19 (-0.53, 0.98)

Seson Length 0.51 (-0.1, 1.24)

Rainfall -1.48 (-2.48, -0.004)

Maximum Temperature -1.31 (-2.38, 0.73)

Maximum Temperature 2 -0.03 (-2.27, -0.02)

Minimum Temperature 1.80 (-2.18, 2.71)

Minimum Temperature 2 0.05 (-0.03, 0.1)

Distance to water 1.08 (0.46, 2.51)

Distance to water 2 -0.29 (-0.56, 0.09)


Table 5.1: Posterior estimates for environmental covariate effects.

Figure 5.2: Posterior distribution of the number of tiles.

5.5 Analysis of the malaria prevalence data 93

Figure 5.3: Percentiles (2.5th, 50th and 97.5th) of the posterior distribution for σ2 and ρ.

Predictions of malaria risk at 60, 000 unsampled locations covering the whole area of sub-

Sahara Mali were carried out using Bayesian kriging. The map of malaria prevalence is

shown in Figure 5.4. High malaria prevalence was predicted in the west part of the country

(Kayes region), a small area in the center of Mali (Koulikoro and Segou regions) and in

the south-east part of Mali. Low levels of malaria risk were predicted in the north part of

the country, the region at the border with Mauritania and the center of Koulikoro region.

The prediction error is depicted in Figure 5.5. We observe that predictions have lower

variances in areas around the data locations and the prediction error is higher in regions

remote from the sampling locations.


Figure 5.4: Map of predicted malaria prevalence for sub-Sahara Mali, estimated from themedian of the posterior predictive distribution.

Figure 5.5: Map of prediction error for sub-Sahara Mali, estimated from the standarddeviation of the posterior predictive distribution.

5.6 Discussion 95

5.6 Discussion

In this paper we extended our earlier work (Gosoniu et al., 2006) which considered the non-

stationary feature of malaria by dividing the area of interest in fixed number of regions,

assumed a separate stationary spatial process in each region and correlation between the

regions. That approach is more appropriate when the data analyzed cover a large area with

a fixed space partitioning such as ecological zones defined by precipitation, evaporation

and availability of water. Non-stationarity may also be explained by factors other than

the environmental ones (socio-economic status, human activities etc.) which may influence

the spatial correlation differently over the study area. However, in practical situations a

fixed partitioning may not be obvious. In the present approach we let the statistical model

capture the different spatial processes by allowing the number and the configuration of the

regions to be random. The model parameter space has variable dimensions, depending on

the tessellation, therefore we used RJMCMC for model fit. A random tessellation model

has been employed also by Gemperli (2003) for mapping malaria in Mali. However, the

authors assumed independence of the data across the regions. This assumption implies that

neighboring points located in different tiles are not correlated. Our contribution was to

develop a random tessellation model which takes into account the between tile correlation.

In Gosoniu et al. (2006) the model validation results between a stationary and a fixed

tile-based non-stationary model were reported, showing that malaria mapping is sensitive

to these model assumptions. Ignoring the non-stationarity characteristic may lead to unre-

liable estimates of the risk factors’ effects on malaria and to inaccurate prediction estimates

of malaria prevalence. In addition, the spatial correlation gives an indication of the impor-

tance of geographically structured factors as well as of the unmeasured local factors, such

as human behavior.

The malaria map based on the non-stationary model developed in this paper shows over-

all, similar risk patterns as the previous two maps that allowed for non-stationary spatial

covariance structure (Gemperli, 2003; Gosoniu et al., 2006), as well as significant differ-

ences. In particular, all three approaches estimated high malaria risk (> 0.7) in the region

of Kayes and low prevalence (< 0.2) in the region of Tombouctou. However, our map

indicated larger areas with high parasitaemia risk, as well as certain areas with smaller

risk compared with the other two approaches. In particular, in the south of Koulikoro

region and the west of Mopti region we predicted higher parasitaemia risk that the other

two methods. Similarly, at the border with Burkina Faso we predicted malaria prevalence


higher than 0.7 in our case, while Gemperli (2003) and Gosoniu et al. (2006) estimated

a prevalence of 0.4 − 0.7. Our map shows lower level of malaria prevalence at the border

with Mauritania and in the region of Gao compared with the map based on the approach

developed by Gemperli (2003). The malaria risk map produced by the current model was

validated by expert opinion who suggested that the estimates obtained reflect better the

malaria situation on the ground. The maps indicate that the assumptions involved in

modeling non-stationarity influence the resulting maps.

Previous approaches that modeled non-stationarity (Gemperli, 2003; Kim et al., 2005)

assumed independence between neighboring points located in different tiles. We ad-

dressed this issue by considering correlation between the random tiles. Although our

model methodologically improves the previous modeling approaches, it has larger number

of parameters because at each location it includes as many random effects as the number of

tiles. Further research on simulated data is required to show if the more complex model is

able to capture better the spatial processes over the study region than the simpler model,

with fewer parameters but stronger assumptions.

Fitting geostatistical models for non-Gaussian data involves repeated inversion of the spa-

tial covariance matrices and, for large number of locations the operation becomes com-

putationally intensive. The previous methods that assumed tile-independence have the

advantage that facilitates the matrix inversion by converting the spatial covariance matrix

into block diagonal form. Our model requires inversion of the spatial covariance matrix as

many times as the number of tiles. To overcome this computational challenge one could

approximate the tile-specific spatial processes by sparse Gaussian processes (Seeger et al.,

2003; Snelson and Ghahramani, 2006), estimating the spatial processes from a subset of

m << n locations within each tile. In this way the problem would be reduced to inversion

of much smaller matrices mxm, instead of the original nxn matrices.

We are currently further developing this approach by assuming that the relation between

the environmental factors and the malaria risk is tile specific.

Acknowledgments

The authors would like to thank the MARA collaboration for making the malaria data

available. This work was supported by the Swiss National Foundation grant Nr.3252B0-

102136/1.

Chapter 6

Mapping malaria using mathematical

transmission models

Gosoniu L.1 ,Vounatsou P.1, Riedel N.1, Sogoba N.2, Miller J.3,Steketee R.3, Smith T.1

1 Swiss Tropical Institute, Basel, Switzerland2 Malaria Research and Training Center, Universite du Mali, Bamako, Mali3 The Malaria Control and Evaluation Partnership in Africa, Zambia

This paper is being prepared for submission to American Journal of Epidemiology.

98 Chapter 6. Mapping malaria using mathematical transmission models

Summary

Historical malaria field survey data used for mapping the geographical distribution of the

disease have a number of limitations that make their analyzes challenging. The surveys

are carried out at different seasons with non-standardized and overlapping age groups of

the population. To overcome these problems, we propose the use of a newly developed

mathematical malaria transmission model to translate the heterogeneous age prevalence

data into a common measure of transmission intensity like entomological inoculation rate.

This approach was applied on malaria data extracted from the Mapping Malaria Risk

in Africa (MARA) database for Mali. A Bayesian geostatistical model was fitted to the

estimates of transmission intensity and using Bayesian kriging we produced a smooth

map of annual entomological inoculation rates in Mali. This map was converted to age

specific malaria risk maps using again the transmission model. Model validation revealed

that the geostatistical model based on the estimates derived from the transmission model

had a better predictive ability compared to the one modeling the raw prevalence data.

To further assess the malaria risk maps based on the transmission model, data from the

nationwide Malaria Indicator Survey (MIS) in Zambia were analyzed. Maps based on

both the transmission model and the raw prevalences were compared. Results showed

that the map based on the transmission model predicts similar patterns of malaria risk

with the one obtained by analyzing directly the MIS data. The proposed approach in

malaria mapping has the advantages that i) age and seasonality adjusted malaria risk

maps can be obtained from prevalence data compiled from different sources ii) all survey

data can be used in mapping despite the age-heterogeneity and iii) maps of different malaria

transmission measures can be produced from survey data.

Keywords : Bayesian kriging, entomological inoculation rate, malaria mapping, Markov

chain Monte Carlo, parasite prevalence.

6.1 Introduction 99

6.1 Introduction

Mapping the geographical patterns of malaria risk and estimates of population at risk are

important for planning, implementation and evaluation of malaria control programs. They

provide important information for optimizing the resource allocation for malaria control

in high risk areas. The national and global estimates of burden of disease are imprecise

because of the inadequate malaria case reporting in most endemic countries as well as the

lack of national wide malaria surveys. Consequently there is a renewed effort in mapping

malaria risk (Hay et al., 2006; WHO, 2007) with the aim of producing baseline maps of

malaria transmission.

Most empirical maps of malaria in sub-Sahara Africa are based on field survey data on

prevalence of infection. Todate the main sources for these data are published and un-

published reports. The compiled databases have a number of problems which complicate

their analyzes. First, the surveys are carried out in different seasons,hence it is difficult to

account for seasonality in modeling malaria transmission. Second, the population covered

by different surveys has non-standardized and overlapping age groups, therefore adjusting

for age could be challenging.

The most complete database on malariometric data across Africa is the ”Mapping Ma-

laria Risk in Africa” (MARA/ARMA, 1998). It contains over 10, 000 distinct age-specific

prevalence values since the early 1960s. The advantage of analyzing historical data, in

particular data extracted from the MARA database is that they provide a wealth of infor-

mation in assessing temporal changes of malaria. However, the derived malaria maps may

not indicate the actual situation of malaria at a specific location, which could be affected

by control interventions or human activities (e.g. irrigation, dams). Most of the analysis

done so far on MARA data (Kleinschmidt et al., 2001; Gemperli et al, 2003; Gosoniu et al.,

2006) were concentrated on a specific age group, discarding surveys with overlapping age

groups of the population. In this way much of the data are unused, resulting in unreliable

malaria transmission estimates in areas with sparse data.

A new source of malaria data is Malaria Indicator Surveys (MIS), developed by Roll Back

Malaria (RBM) in 2004 for monitoring coverage of malaria prevention and treatment. The

MIS package is a stand-alone survey tool that can be used to collect malaria-related data

in countries that are lacking such data for malaria program management. The Zambia

National MIS is the first nationally representative household survey assessing coverage of

malaria interventions and malaria-related burden in children under 5 years of age during


May-June 2006. Similar surveys have been conducted in Angola, Mozambique, Ethiopia

and Senegal.

To overcome the problems related to the MARA database, Gemperli et al. (2005; 2006)

employed the Garki malaria transmission model (Dietz et al, 1974) to convert observed

prevalence data into an estimated age-independent entomological measure of transmission

intensity, which was further used for mapping purposes. This work demonstrated the fea-

sibility of using malaria transmission models in malaria risk mapping. However, the Garki

model was developed on field data from the savanna zone of Nigeria, therefore it cannot be

generalized to other regions in Africa with different environmental conditions and levels of

malaria endemicity. Recently, the modeling group of the Swiss Tropical Institute developed

a new malaria transmission model (Smith et al., 2006) which overcomes a number of limi-

tations of the Garki model. This is an individual-based stochastic model which simulates

age-specific malaria epidemiological outcomes (i.e. parasitaemia, morbidity, mortality) at

a given location, conditional on the seasonal pattern in the entomological inoculation rate

(EIR). In addition, a seasonality model has been developed (Mabaso, 2007) to estimate

from climatic factors monthly EIR due to Anopheles gambiae s.l. at any location in Africa.

We employed this mathematical model to map malaria survey data from Mali, adjusting

for age and seasonality. In particular, we translated the observed MARA prevalence data

into estimates of EIR. Using environmental and climatic data from remote sensing (RS) as

predictors we fitted a Bayesian geostatistical model on the estimates of EIR. We further

employed Bayesian kriging to obtain a smooth map of EIR for Mali. Applying again

the mathematical model we converted the predicted EIR values into estimates of malaria

prevalence for children less than 5 years old. We assessed the predictive ability of the

geostatistical intensity model in malaria mapping and compare it with the model that

analyzed directly the raw prevalence data.

To further assess the malaria risk maps based on the transmission model, we analyzed the

Zambia MIS data by employing the transmission model as well as modeling directly the

prevalence data and compare the resulting maps. The Zambia MIS data were previously

analyzed by Riedel et al. (unpublished) by fitting Bayesian geostatistical models. Bayesian

P-splines (Eilers and Marx, 1996) were employed within a geostatistical model to take into

account the non-linear relation between parasitaemia risk and environmental factors in the

analyzes of Mali and Zambia data.

6.2 Materials and methods 101

6.2 Materials and methods

6.2.1 Malaria data

Malaria prevalence data for Mali were extracted from the MARA/ARMA database. We

analyzed data from 497 surveys carried out at 115 distinct locations between 1962 and

2001. The surveys covered different age groups of the population (Table 6.1). During

these surveys, 104, 689 individuals were examined and a proportion of 46.67% were found

with P. falciparum parasites in the blood sample.

Age groups (years) Nb. of surveys Age groups (years) Nb. of surveys

0-9 13 6-99 1

0-12 6 14-32 1

0-15 3 15-99 1

0-44 9 0-5 & 6-10 4

0-99 1 0-9 & 0-20 3

1-1 1 2-4 & 5-9 17

1-9 2 2-9 & 10-99 22

1-12 2 5-9 & 10-14 30

1-15 1 8-14 & 15-19 8

1-70 1 0-1 & 2-4 & 5-9 9

2-9 47 1-2 & 3-5 & 6-10 30

2-15 3 1-15 & 5-9 7

& 6-14

5-9 6 2-4 & 5-9 30

& 10-14

5-15 1 2-4 & 5-9 33

& 10-15

6-14 3 1-1 & 2-4 180

& 5-9 & 10-14

6-18 4 0-4 & 0-12 & 2-9 14

& 5-9 & 10-18

6-20 1 1-4 & 2-4 & 5-9 8

& 10-14 & 15-19 & 15-24

Table 6.1: Age groups of the population included in the MARA surveys in Mali betweenyears 1962-2001.


Data on malaria prevalence in Zambia are available from the Zambia National Malaria

Indicator Survey conducted during May-June 2006. The sample frame of the survey was the

list of 17, 000 Standard Enumeration Areas (SEAs) from the 2000 Population Census. The

survey was carried out on a sample of 120 SEAs with 25 households in every SEA, including

2, 364 children under 5 years of age. Malaria prevalence data were collected by finger prick

blood sampling in children under five. The coordinates of the households were recorded

using personal digital assistants (PDAs). The data were aggregated at SEA level and we

calculated the coordinates of SEAs as the mean of the households coordinates belonging

to each SEA. After eliminating the children with incomplete information, the final dataset

analyzed here was collected at 109 distinct SEAs and comprised 1, 324 children.

6.2.2 Environmental data

In the analyzes of malaria data in Mali we used the following environmental and climatic

variables: normalized difference vegetation index (NDVI), soil water storage index (SWS),

rainfall, temperature, distance to the nearest water body and land use. The predictors

used in the analyzes of MIS data from Zambia were: NDVI, rainfall, temperature, potential

evapotranspiration (PET) and land use. The climatic and environmental data used in this

analysis were obtained from different sources and at different resolutions for both Mali and

Zambia (Table 6.2 and Table 6.3).

The land cover data downloaded for Mali included 24 categories. We regrouped them

in the following 3 classes: urban/ dry/ barren/ sparsely vegetated (land use 1), crop/

grassland/ mosaic(land use 2) and wet/ irrigated crop land/ savanna (land use 3). We

created a two kilometers buffer area around each data point and calculated the relative

frequencies of the land use classes in each buffer. Every land use class is considered as a

separate variable with values between 0 and 1. In Zambia we obtained the land cover in

16 categories which we regrouped into 5 broad classes, namely: wet area, forest, urban,

shrub land and other. Similarly to Mali, we created for every land use class a variable

corresponding to the relative frequency of the pixels of the different land use categories

inside a two kilometers buffer.


Factor Resolution Source

Normalized Difference Vegetation Index (NDVI) 8km2 NASA AVHRR Land data sets

Soil Water Storage Index (SWS) 5km2 Droogers et al., 2001

Rainfall 5km2 Hutchinson et al., 1996

Temperature 5km2 Hutchinson et al., 1996

Water bodies 1km2 World Resources Institute, 1995

Land use 1km2 USGS-NASA

Table 6.2: Spatial databases used in the spatial analysis in Mali.

Factor Resolution Source

Normalized Difference Vegetation Index (NDVI) 250m2 MODIS

Rainfall 8km2 ADDS

Temperature 1km2 MODIS

Evapotranspiration (PET) 8km2 ADDS

Land use 1km2 MODIS

Table 6.3: Spatial databases used in the spatial analysis in Zambia.

6.2.3 Statistical analysis

Malaria transmission models

The transmission model of Smith et al. (2006) simulates malaria epidemiological outcomes

(i.e. parasitaemia, morbidity, mortality) at a specific location, conditional on the seasonal

pattern in the EIR. Treating EIR as a known quantity, but allowing for temporal variations

and the influence of age host, infections are introduced into a population of simulated

humans via a stochastic process dependent on the EIR and then sample the subsequent

parasite densities using 5-day time steps. Multiple field datasets across Africa have been

used to optimize parameter estimates. A detailed description of the model is given in

Smith et al. (2006) and Smith et al. (unpublished).


Geostatistical models

Fitting malaria transmission intensity data

A Bayesian geostatistical model was fitted on the Box Cox transformed EIR values Yi esti-

mated from the transmission model at each location i, i = 1, . . . , n. We assume that Yi are

normally distributed, Yi ∼ N(µi, τ2) and introduce the environmental covariates on the

mean structure µi. To overcome the non-linearity in the relation between environmental

predictors and malaria transmission we employed a non-parametric regression model using

P-splines. Details on the implementation of this approach are given in Chapter 4. In brief,

µi is modeled as µi =∑p

j=1 fj(Xij) + φi, where f(·) is a smooth function of the environ-

mental predictors, that is fj(Xij) =∑K

k=1 ukj|Xij − skj|3, where uj = (u1j, . . . , uKj)T is

the vector of regression coefficients, s1j < s2j < . . . < sKj are fixed knots and |Xij − skj|3

is a truncated 3-rd order polynomial spline basis, all related to covariate Xj. We consider

the knots to be based on sample quantiles of the covariates.

The parameters φi are location-specific random effects modeling the spatial correlation by

assuming that they derive from a multivariate normal distribution φ = (φ1, φ2, . . . , φn)T ∼MV N(0, Σ) with the variance-covariance function related to an exponential correlation

function between locations. In particular, Σij = σ2exp(−dijρ), where dij is the Euclidean

distance between locations i and j, σ2 captures within locations spatial variation and is

known as the sill and ρ is called the decay parameter and measures the rate of decrease

of correlation with distance. In this exponential setting, the decay parameter is translated

into the range parameter, that is the minimum distance for which the spatial correlation

is less than 5%, by calculationg 3/ρ. Markov chain Monte Carlo (MCMC) simulation

techniques were employed to estimate the model parameters. We used Bayesian kriging to

predict Box Cox transformed EIR values at locations where malaria data are not available

and produce smooth maps of EIR in Mali and Zambia.

We applied the relations estimated by the malaria transmission model and transformed

back the predicted EIR values to age related malaria prevalence. We produced maps of

malaria risk for children under 5 years old and 1 − 10 years old for Mali and only for

children under 5 in Zambia. The software used in this analysis was written by the authors

in Fortran 95 (Compaq Visual Fortran Professional 6.6.0) using standard numerical libraries

(NAG, The Numerical Algorithms Group Ltd.). Further details on the Bayesian modeling

approach are given in the Appendix of this chapter.


Fitting malaria prevalence data

We also analyzed directly the malaria survey data from Zambia, by fitting a Bayesian

geostatistical model, using as predictors the environmental factors mentioned in Section

6.2.2. Let Ni be the number of children screened at location i and Yi the number of

children found positive to P. falciparum parasitaemia. We assume that Yi arises from

a Binomial distribution, that is Yi ∼ Bin(Ni, pi) with parameter pi measuring malaria

risk at location i. Similarly to the previous section we assume a non-linear environment-

malaria relation and model it using P-splines via the logistic regression, that is logit(pi) =∑pj=1 fj(Xij) + φi. The spatial dependency is modeled on the covariance matrix of the

location-specific random effects. We assumed that the covariance Cov(φi, φj) between

every pair (φi, φj) decreases with their distance dij and, like in the previous geostatistical

model, we choose an exponential correlation function. More details of this approach are

given in the Appendix of this chapter.

Model validation

Using the malaria dataset from Mali, we assessed the predictive abilities of both the model

that analyzed directly the raw prevalence data as well as the one that models transmission

intensity data derived from the mathematical model. Surveys carried out in children

between 1−10 years old were selected to model directly the prevalence data. Model fit was

implemented in 85% of the survey locations and validation was performed in the remaining

ones. The geostatistical model based on the estimates derived from the transmission model

was fitted on the entire MARA dataset, except for the test locations used for validation.

Following the approaches developed in Gosoniu et al. (2006), the predictive ability of the

two models was assessed using Bayesian ”p-values” and the probability coverage of the

shortest credible interval. In particular, for each test location we calculated the area of the

predictive posterior distribution which is more extreme than the observed data. The model

predicts well the observed data when the median of the posterior predictive distribution

is close to the observed data. We assert that the model with the best predictive ability is

the one with the ”p-value” closer to 0.5. In addition we calculated 12 credible intervals of

the predictive posterior distribution at the test locations with probability coverage equal

to 5 %, 10 %, 20 %, 25 %, 30 %, 40 %, 50 %, 60 %,70 %, 80 %, 90 % and 95 %, respectively

and compared the proportions of test locations with observed malaria prevalence falling in

the above intervals.


6.3 Results

The Mali MARA dataset contained 330 surveys conducted in children between 1−10 years

old at 87 distinct locations. The geostatistical model based on the raw prevalence data was

fitted on randomly selected 317 surveys at 74 distinct locations (training set). Surveys at

the remaining 13 locations were used for validation (test points). The geostatistical model

based on the transmission intensity data was fitted on 476 surveys over 102 locations, that

is on the entire MARA data set except the 13 test locations.

Each box-plot in Figure 6.1 summarizes the distribution of the 13 Bayesian ”p-values”

calculated from the predictive posterior distribution of the test locations for the model

that directly analyzed the prevalence data (left) and for the approach that modeled the

transmission intensity data derived from the mathematical malaria transmission model

(right). The median of this distribution for the latter model in closer to 0.5, suggesting

that this is the best model.

Figure 6.1: The distribution of Bayesian p-values. The box plots display the minimum,the 25th, 50th, 75th and the maximum of the distribution.

6.3 Results 107

Figure 6.2 shows the percentages of test locations with malaria prevalence falling in each

of the 12 credible intervals of the posterior predictive distribution for both geostatistical

models. We observe that the wider credible interval (95%) includes the same percentage

of test locations (69.23%) for both models, but for the credible intervals ranging from 50%

to 90% the approach based on the transmission model include, consistently, the highest

percentage of observed locations.

Figure 6.2: Percentage of test locations with malaria prevalence falling in the 5%, 10%,20%, 25%, 30%, 40%, 50%, 60%, 70%, 80%, 90% and 95% credible intervals of the posteriorpredictive distribution.

The nonlinear effects of the environmental factors on malaria transmission in Mali and

Zambia are shown in Figure 6.3 and Figure 6.4, respectively. The graphs show the poste-

rior median and the 95% credible intervals. In Mali, the credible intervals of the posterior

medians for all the variables contain 0, therefore none environmental factor was signif-

icantly associated with annual EIR. In Zambia, the only credible interval that did not

include zero was the one corresponding to the posterior median of the evapotranspiration,


therefore we conclude that this was the only variable accounting for significant spatial

variation in malaria transmission in this dataset. We notice a slightly decreasing effect of

evapotranspiration on annual EIR.

Figure 6.3: Estimated effect (P-spline) of environmental factors on EIR in Mali. Theposterior median (pink) and the 95% credible interval are shown.

Figure 6.4: Estimated effect (P-spline) of environmental factors on EIR in Zambia. Theposterior median (pink) and the 95% credible interval are shown.

6.3 Results 109

Table 6.4 shows the posterior estimates of the spatial parameters: decay parameter, spatial

variance and residual non-spatial variance. In Mali, the posterior median of the decay

parameter ρ was equal to 2.60 km (95% credible interval: 0.73, 8.01), which translates to

a minimum distance where correlation drops below 5% of around 1.15 km (95% credible

interval: 0.37, 4.14). The posterior distribution of ρ had the median equal to 0.11 km

(95% credible interval: 0.04, 0.59) in Zambia, corresponding to a range of 28.52 km (95%

credible interval: 5.10, 68.45). Estimates of the range parameters suggest a weak spatial

correlation in Mali and a strong spatial correlation in Zambia. In Mali, the spatial variation

was slightly smaller (σ2 = 1.74) than the residual non-spatial variation (τ 2 = 2.62). The

opposite situation was in Zambia, where the spatial variation was very high (σ2 = 7.23)

compared to the non-spatial variation (τ 2 = 0.11).

Country Spatial parameter Median 95% CIa

Mali ρ 2.60 (0.73, 8.01)

σ2 1.74 (0.98, 2.92)

τ 2 2.62 (2.27, 3.03)

Zambia ρ 0.11 (0.04, 0.59)

σ2 7.23 (2.84, 10.90)

τ 2 0.11 (0.01, 3.95)


Table 6.4: Posterior estimates of spatial parameters.

The smooth map of predicted annual EIR in sub-Saharan Mali is shown in Figure 6.5.

It depicts high malaria transmission in the region of Kayes (west of Mali), in the central

part of Gao region and in the east of Mopti region (at the border with Burkina Faso).

Low transmission of malaria is predicted at the border with Mauritania, east side of region

Mopti, south of Gao region and some parts in the region of Sikasso (south of Mali). The

corresponding prediction error is shown in Figure 6.6. The error is very low near the

sampling locations and it increases toward the north of Mali, where malaria surveys are

very sparse.

We employed the malaria transmission model and converted the predicted EIR values to

malaria prevalence for children under 5 years old and for children 1 to 10 years old. The

two malaria risk maps for Mali are shown in Figure 6.7 and Figure 6.8. Both maps depict


a pattern similar with the predicted annual EIR map, with malaria risk for children 1 to

10 years old uniformly higher than for children under 5 years old.

Figure 6.5: Predicted annual entomological inoculation rate (EIR) in Mali estimated fromthe median of the posterior predictive distribution.

Figure 6.6: Prediction error of annual entomological inoculation rate (EIR) in Mali es-timated from the standard deviation of the posterior predictive distribution. Samplinglocations are indicated by circles.

6.3 Results 111

Figure 6.7: Predicted malaria prevalence for children under 5 years old in Mali estimatedfrom the median of the posterior predictive distribution.

Figure 6.8: Predicted malaria prevalence for children between 1 and 10 years old in Maliestimated from the median of the posterior predictive distribution.


Figure 6.9 depicts the predicted annual EIR in Zambia. Overall, the model predicts low

level of malaria transmission. The highest level of transmission is observed in the eastern

part of the country and some regions in the northern part. The lowest malaria transmission

was predicted in the north-west of Zambia and a small area in the south of the country.

Figure 6.10 presents the prediction error of Zambia model. We observe higher error in the

north-west and southern part of the country, at locations remote from the survey locations

and low prediction error in areas around the sampling locations.

Figure 6.9: Predicted annual entomological inoculation rate (EIR) in Zambia estimatedfrom the median of the posterior predictive distribution.

6.3 Results 113

Figure 6.10: Prediction error of annual entomological inoculation rate (EIR) in Zambiaestimated from the standard deviation of the posterior predictive distribution. Samplinglocations are indicated by circles.

A smooth map of malaria risk for children under 5 years in Zambia is shown in Figure

6.11. Similar to the map of transmission intensity, high malaria risk is predicted in the

eastern part and some regions in the north of Zambia, as well as at the border with Zim-

babwe. Figure 6.12 depicts the smooth map of parasitaemia risk for children under 5 years

old in Zambia obtained by fitting the logistic geostatistical model on the malaria survey

data. This map shows a similar pattern of malaria risk to the one obtained by employing


the transmission model. However, the former map predicts lower level of parasitaemia

prevalence (< 0.2) compared with the latter one (0.2-0.4). We calculated the Kendall’s

correlation coefficient to measure the degree of correspondence between parasitaemia risk

estimated by the mathematical model and by directly analyzing the prevalence data. We

obtained Kendall’s tau equal to 0.22 and p-value < 0.001, indicating that the two measures

are statistically significant associated.

Figure 6.11: Predicted malaria prevalence for children under 5 years old in Zambia esti-mated from the median of the posterior predictive distribution.

6.3 Results 115

Figure 6.12: Predicted malaria prevalence for children under 5 years old in Zambia esti-mated from the median of the posterior predictive distribution obtained by directly ana-lyzing the prevalence data.


6.4 Discussion

In this paper, a newly developed malaria transmission model was employed to produce

age-adjusted malaria risk maps from survey data extracted from different sources. The

age-heterogeneous prevalence data were converted into an age independent measure of

transmission, the so-called entomological inoculation rate (EIR). Smooth maps of EIR

were converted to age-specific malaria risk maps, using the relation between prevalence

and transmission intensity described by the mathematical transmission model for each age

group.

This approach was applied to analyze MARA survey data from Mali. Model validation

revealed that the geostatistical intensity model has a better predictive ability in mapping

the MARA survey data than the geostatistical prevalence model. To further assess the ma-

laria risk maps based on the transmission model, data from the nationwide MIS in Zambia

were analyzed. Maps based on both the transmission model and the raw prevalences were

compared. Results showed that the map based on the transmission model predicts similar

patterns of malaria risk with the one obtained by analyzing directly the MIS data. The

advantage of using the transmission model in malaria mapping is that it makes possible

age and seasonality adjustment as well as the inclusion of all available malaria historical

data collected at different age groups. In addition, maps of different malaria transmission

measures can be produced from survey data.

The Bayesian P-spline regression approach models the non-linear relation between envi-

ronmental/climatic factors and malaria transmission in a flexible way. In Mali, none of

the environmental variables used in the analyzes were statistically significant associated

with annual EIR. In Zambia the model estimates a slightly negative association between

evapotranspiration and malaria transmission. The spatial correlation present in the data

was modeled in a Bayesian framework, using MCMC simulation techniques which enables

simultaneously estimation of all model parameters together with their standard errors. We

assessed the precision of the smooth maps of annual EIR by quantifying the prediction

error using Bayesian kriging.

Gemperli et al. (2005) made use of the Garki malaria transmission model and produced

maps of malaria transmission intensity and malaria risk for Mali. However, the Garki

model was developed using field data from the savanna zone of Nigeria, therefore it should

not be generalized to other regions in Africa with different environmental conditions and

malaria endemicity. In addition, they ignored the seasonal patterns of malaria transmission,

6.4 Discussion 117

whereas we used the seasonality model developed by Mabaso et al. (2007) and adjust for

season at which data were collected. Comparing the map of annual EIR estimated by

our model with the map obtained by Gemperli et al. (2005) we observe that overall we

predicted lower level of EIR. This difference could be explained by the fact that Gemperli

et al. (2005) did not adjust for seasonality in transmission. The malaria risk maps for Mali

(for the two age groups) are similar with the maps obtained from previous work (Gosoniu

et al., 2006).

The maps produced for Zambia indicate low level of malaria transmission intensity. This

could be related to the reduced amount of rainfall experienced by Zambia (2.0− 3.5 mm)

and to the recent malaria control intervention implemented in the country. The only recent

available data on EIR in Zambia are given by Kent et al. (2007) who conducted a study

in two villages in the Southern Province of Zambia during November 2004-May 2005 and

November 2005-May 2006 to investigate the seasonal intensity of malaria transmission in

this region. During the first period malaria transmission was undetectable because of severe

drought experienced by Zambia, but in the next season EIR was estimated at 1.6 and 18.3

infective bites per person per transmission season in the two villages, respectively.

The modeling approach used in this study was based on the assumption of stationarity, that

is the spatial correlation was considered a function of only the distance between locations

and irrespective of locations themselves. Malaria has non-stationary feature since local

characteristics (environment, land use, vector ecology) influence the spatial correlation

differently at various locations. Gosoniu et al. (2006) showed the importance of statistical

modeling approach when analyzing malaria data. The methodology presented here could

be extended to take into account the non-stationary characteristic of malaria by using the

Bayesian partition modeling approach for geostatistical data developed in Chapter 5.

The P-spline approach which was used to model the non-linear effect of environmental

conditions could be further developed by allowing the model to choose the number and the

position of the knots instead of the fixed knots based on sample quantiles of the covariates

used in this study. These models would have a variable number of parameters, therefore

fitting would require the use of Reversible Jump MCMC (RJMCMC) (Green et al., 1995).

Acknowledgments

This work was supported by the Swiss National Foundation grant Nr.3252B0-102136/1.


6.5 Appendix

Geostatistical model of transmission intensity

Let Yi be the Box Cox transformation of the annual EIR at location i, i = 1, . . . , n. We

assume that Yi are normally distributed, Yi ∼ N(µi, τ2) with the mean structure µi a

function of the environmental factors and τ 2 the unexplained non-spatial variance. The

non-linear effect of the environmental predictors is modeled using Bayesian P-splines, that

is µi =∑p

j=1 fj(Xij) + φi, where f(·) is a smooth function of the environmental variables.

In particular,

fj(Xij) =∑K

k=1 ukj|Xij − skj|3,

where uj = (u1j, . . . , uKj)T is the vector of regression coefficients, s1j < s2j < . . . < sKj

are fixed knots and |Xij − skj|3 is a truncated 3-rd order polynomial spline basis, all

corresponding to covariate Xj. The knots were chosen to be based on sample quantiles of

the covariates.

Spatial correlation was modeled by assuming that the random effects φi introduced at each

location i are distributed according to a multivariate normal distribution

φ = (φ1, φ2, . . . , φn)T ∼ MV N(0, Σ), where Σij is a parametric function of the distance dij

between locations i and j. A commonly used parametrization is Σij = σ2corr(dij; ρ), where

σ2 is the spatial variance and corr(dij; ρ) is a valid correlation function. For the current

application we chose the exponential correlation function corr(dij; ρ) = exp(−dijρ), where

the parameter ρ captures the scale of correlation decay with distance. Ecker and Gelfand

(1997) proposed several other parametric correlation forms, such as Gaussian, Cauchy and

spherical.

Following the Bayesian modeling specification, we need to adopt prior distributions for all

the parameters in the model. We adopt non-informative normal prior for the regression

coefficients p(u) ∼ N(0, 100) and the following priors for σ2, τ 2 and ρ: p(σ2) ∼ Inverse

Gamma(a1, b1), p(τ 2) ∼ Inverse Gamma(a2, b2) and p(ρ) ∼ Gamma(a3, b3) with the hyper-

parameters a1, b1, a2, b2 and a3, b3 chosen to obtain a prior mean equal to 1 and a variance

equal to 100. Bayesian inference is based on the joint posterior distribution

p(u, φ, σ2, τ 2, ρ|Y ) = L(u, φ; Y )p(u)p(φ|σ2, ρ)p(σ2)p(τ 2)p(ρ),

where L(u, φ; Y ) is the likelihood function and p(φ|σ2, ρ) is the distribution of the random

effects, p(φ|σ2, ρ) ∼ MV N(0, Σ).

6.5 Appendix 119

The model parameters were estimated using Markov chain Monte Carlo (MCMC) simu-

lation and in particular Gibbs sampling. The conditional distribution of the regression

coefficients is conjugate normal from which we can easily draw samples. The remaining

parameters do not have conditional distributions of standard form and we employed a

random walk Metropolis algorithm for sampling. We adopt a Gaussian proposal for the

random effects φ and a Gamma proposal for σ2, τ 2 and ρ. The mean of the proposal

distribution was the parameter estimate from the previous iteration and the variance was

adaptively adjusted to achieve an acceptance rate of about 0.4.

Prediction of the Box Cox transformations of annual EIR at locations were malaria data

are not available was carried out using Bayesian kriging. Let Y 0 = (Y 01 , . . . , Y 0

m)T be the

values to predict at the new m locations. Given the random effects φ at the observed

locations, we predict the random effects φ0 at the new locations by sampling from the

normal distribution

P (φ0|φ, σ2, ρ) = N(Σ01Σ−111 φ, Σ00 − Σ01Σ

−111 ΣT

01),

where Σ11 = E(φφT ), Σ00 = E(φ0φ0T ) and Σ01 = E(φ0φT ).

The predictive distribution is

P (Y 0|Y ) =∫

P (Y 0|u, φ0)P (φ0|φ, σ2, ρ)P (u, φ, σ2, τ 2, ρ|Y ) du φ0 φ dσ2 dτ 2 dρ,

where P (u, φ, σ2, τ 2, ρ|Y ) is the posterior distribution. Conditional on u and φ0i , Y 0

i is

drawn from the normal distribution N(∑p

j=1 fj(X0ij) + φ0

i , τ2), where X0

i = (X01 , . . . , X

0p )T

are the environmental covariates at the new location. Bayesian spatial prediction is per-

formed by consecutive drawing samples from the posterior distribution, the distribution of

the new random effects and the normal distribution of the predicted outcome.

Geostatistical model of prevalence

Let Yi be the number of observed malaria cases out of Ni children examined at location

i, i = 1, . . . , n and Xi = (Xi1, Xi2, . . . , Xip)T be the vector of p associated environmental

predictors observed at location i. We assume that Yi are binomially distributed, that is Yi ∼Bin(Ni, pi) with parameter pi measuring malaria risk at location i. The non-linear effect

of environmental conditions is modeled non-parametrically on the logit transformation of

pi, that is logit(pi) =∑p

j=1 fj(Xij) + φi. The smooth function f(·) is defined as in the

intensity geostatistical model.

The spatial correlation is modeled on the covariance matrix of the location-specific random


effects, that is φ = (φ1, φ2, . . . , φn)T ∼ MV N(0, Σ), where Σij = σ2exp(−dijρ). The

spatial covariance parameters are described in the previous section, as well as their prior

distributions. Model fit is handled via MCMC simulations. The conditional distribution

of the spatial variance σ2 is conjugate Gamma and we use Gibbs sampler to estimate the

parameter. For the remaining parameters we employed Metropolis algorithm for sampling,

since their conditional distributions have non-standard forms.

We obtain estimates of the malaria risk at any unsampled location by the predictive dis-

tribution

P (Y 0|Y , N ) =∫

P (Y 0|β, φ0)P (φ0|φ, σ2, ρ)P (β, φ, σ2, ρ|Y , N ) dβ dφ0 dφ dσ2 dρ,

where Y 0 = (Y 01 , . . . , Y 0

m)T are the predicted number of cases at new locations,

P (β, φ, σ2, ρ|Y , N ) is the posterior distribution and φ0 = (φ01, . . . , φ

0m)T is the vector of

random effects at new sites. The distribution of φ0 at new locations given φ at observed

locations is normal

P (φ0|φ, σ2, ρ) = N(Σ01Σ−111 φ, Σ00 − Σ01Σ

−111 ΣT

01),

with Σ11 = E(φφT ), Σ00 = E(φ0φ0T ) and Σ01 = E(φ0φT ).

Conditional on φ0i and β, Y 0

i are independent Bernoulli variates Y 0i ∼ Ber(p0

i ) with malaria

prevalence at unsampled site given by logit(p0i ) = X0

i β + φ0i . We predicted malaria pre-

valence at the same unsampled locations where the Box Cox transformation of the annual

EIR was predicted.

Chapter 7

General discussion and conclusions

The epidemiological questions that motivated the work in this thesis were the following: (1)

assessing the spatial effect of bednets use on child mortality; (2) producing different maps of

malaria risk at country and regional level and (3) producing malaria transmission intensity

maps and malaria risk maps adjusted for age and seasonality. These applications led to

the development of novel statistical methods for: (1) modeling large, negative binomial,

geostatistical data; (2) modeling binomial, geostatistical data under the assumption of non-

stationarity; (3) geostatistical models validation and (4) modeling the non-linear effects of

covariates on the outcome variable.

Each chapter in the thesis provides a detailed discussion on the findings. Here is presented

a summary of the main contributions, the implication of our results in malaria control and

some ideas for potential future research.

From methodological point of view, the developed novel statistical methods have several

original contributions in: (i) facilitating estimation of large non-Gaussian geostatistical

models; (ii) estimation and prediction of non-stationary non-Gaussian spatial processes;

(iii) comparing the predictive ability of geostatistical models and (iv) estimation of non-

linear relations between covariates and outcome variables.

In Chapter 2 the spatial effect of bednets use on child mortality was assessed by analyz-

ing data extracted from the Demographic Surveillance System (DSS) set up in Kilombero

Valley, Tanzania. The spatial correlation in mortality data was considered as a function

of the distance between locations and the model was fitted in the Bayesian framework,

122 Chapter 7. General discussion and conclusions

using Markov chain Monte Carlo (MCMC) simulations. The DSS data are collected at

very large number of households, therefore fitting these geostatistical models is computa-

tional intensive because it requires repeated inversions of the variance-covariance matrix.

To overcome this challenge a convolution model for the underlying spatial process was de-

veloped. In particular, the spatial process was estimated by a subset of locations and then

the location-specific random effects were approximated by a weighted sum of the subset of

location-specific random effects with the weights inversely proportional to the separation

distance. This approach has the advantage that it reduces the size of the matrix to be

inverted. There are other methods that overcome very large matrix computations involved

in geostatistical model fit, such as the ones suggested by Rue and Tjelmeland (2002), Stein

et al. (2004), Lee et al. (2005), Xia and Gelfand (2005). An approach similar to the one

developed in this thesis would be the approximation of the spatial process by the projected

latent variables algorithm (PLV) developed by Seeger (2003). Further research may include

validation and comparison of different approaches to accelerate large matrix inversion on

simulated datasets.

The main epidemiological result coming from Chapter 2 is that in an area of high perennial

malaria transmission in southern Tanzania, a community effect of bednets use on all-

cause child mortality was not evident. This result is explained by the small proportion

of insecticide treated nets (ITN), as well as by the homogeneity of bednets coverage in

the study area. To achieve a significant reduction of malaria transmission, hence of child

mortality, the coverage of ITNs and long-lasting insecticidal nets (LLINs) should reach

at least the level of present bednets coverage. In fact, three initiatives namely the Roll

Back Malaria Partnership, the United Nations Millennium Development Goals, and the

US President’s Malaria Initiative have set a target of at least 80% use of ITNs by the

people most vulnerable to malaria (young children and pregnant women) by the year 2010.

Killeen et al. (2007) have shown that a coverage of 35% − 65% of ITN use by the entire

population would have a similar degree of community-wide protection as the personal

protection. Hence, the wide-scale of ITN use by the whole population should be promoted,

keeping still as a priority the use of ITNs by the most vulnerable people.

In Chapter 3 maps of malaria risk in Mali were produced. Maps of malaria distribution are

important tools for increasing the effectiveness of malaria control programs because they

provide useful information on which regions are at high risk of malaria and optimize the

allocation of resources to areas of most need. In addition, the malaria transmission maps

123

can be used to assess the effectiveness of intervention programs. Several maps of malaria

risk have been produced for Mali (Kleinschmidt et al., 2001; Gemperli, 2003; Gemperli et

al., 2006; Sogoba, 2007). Although these maps predicted superficially similar patterns of

malaria transmission, there are significant differences between them because 1) they were

based on different malaria data and environmental predictors and 2) they were produced

using different statistical methods. Spatial correlation in malaria data is likely to be influ-

enced differently at various locations due to local characteristics like environmental factors,

intervention measures, mosquito ecology, health services etc. Therefore, non-stationarity is

an important feature of malaria that can not be ignored when analyzing parasitaemia data.

Maps of malaria risk in Mali under both assumptions of stationarity and non-stationarity

were produced using the same dataset. Non-stationarity was modeled by partitioning the

study area into fixed tiles, assuming a stationary spatial process in each subregion and cor-

relation between the tiles. Comparison of the predictive ability of the two models indicated

that the geostatistical model that allowed for non-stationary spatial covariance structure

predicts better the malaria risk. In addition, the stationarity assumption influenced the

significance of the parasitaemia risk predictors, as well as the estimation of the spatial

parameters. We conclude that malaria mapping is sensitive to the assumptions about the

spatial process. Non-stationarity is an important feature that should be taken into account

in malaria mapping.

The approach that allows modeling non-stationarity, proposed in Chapter 3, is more ap-

propriate when modeling malaria data over large areas with fixed partitions outlined, for

example, by various ecological zones. This methodology was employed in Chapter 4 to

obtain a malaria risk map in West Africa, considering as fixed tiles the four agro-ecological

zones that partition the region. In addition, the non-linear relation between parasitaemia

risk and environmental factors was modeled via Bayesian P-splines, separately in each

agro-ecological zone. Previous malaria maps for West Africa and West and Central Africa

were produced by Kleinschmidt et al. (2001) and Gemperli et al. (2006), respectively, both

under the assumption of stationarity. Kleinschmidt et al. (2001) modeled the non-linear

effects of environmental factors on malaria risk using fractional polynomials and considered

a separate statistical model in each agro-ecological zone. The resulting map showed discon-

tinuities at the edges of the zones, which were further smoothed. However, the modeling

approach of Kleinschmidt et al. (2001) did not allow estimation of the prediction error.

The discontinuities at the borders between the zones were avoided in our case because the

spatial correlation was modeled by a mixture of spatial processes over the entire region,


with the mixing proportions chosen to be exponential functions of the distance between

locations and the centers of the tiles corresponding to each of the spatial processes.

This work, as well as the previous work of Kleinschmidt et al. (2001) was based on

prevalence data collected on specific age groups of the population (less than 10 years

old), while Gemperli et al. (2006) made use of all available data by employing the Garki

transmission model to adjust for age. Gemperli et al. (2006) modeled the non-linear

environment-malaria relation by including interaction terms and using non-linear functional

forms of the predictors (i.e. logarithm, polynomials). However, they assumed the same

relation over the entire region of West and Central Africa. The model developed in Chapter

4 addressed the drawbacks of the previous efforts in mapping malaria in West Africa, by

relaxing the assumptions of stationarity, linearity and the common effect of environmental

conditions on malaria transmission over the different ecological zone. All three maps predict

similar patterns of malaria transmission, but there are also significant discrepancies. The

regions with main differences between the three malaria maps coincide with the areas with

very few or no survey data collected. To improve the current maps, future efforts should

be concentrated on the collection of new data, as well as on the development of novel

statistical models for analyzing the data.

In Chapter 4 the model validation methodology developed in Chapter 3 was further em-

ployed to compare the Bayesian P-spline approach with the method which considers the

covariates as categorical variables. The results indicated that the former model fits better

the non-linear environment-malaria relation. The knots used to define the splines were

fixed, based on sample quantiles of the independent variables. From methodological point

of view, this approach could be extended by allowing the number and the position of the

knots to be random, chosen by the model.

In Chapter 5 a non-stationary model of malaria risk was fitted by assuming a random

rather than a fixed tile configuration, giving a smooth map of parasitaemia risk in Mali.

In fact, the non-stationarity characteristic of malaria data may be explained not only by

the variation in environmental factors, but also by the variation in other factors such

as socio-economic status, malaria control interventions, health care services and human

activities which may influence the spatial correlation differently in various part of the study

area. While the climatic and environmental variables may define a fixed partitioning,

this may not be obvious for the other factors, therefore the data should be allowed to

125

decide on the number of tiles and thus to identify the different spatial processes. Non-

stationarity was modeled by considering the number and configuration of the tiles to be

random parameters in the model. The parameter space has variable dimensions, depending

on the tessellation, hence model fit was handled via a Reversible Jump MCMC sampler.

A random tessellation approach has been developed also by Gemperli (2003) who accounts

for non-stationarity when analyzing malaria prevalence in Mali. However, the author used

a different set of environmental predictors from the ones used in our study and assumed

independence between the tiles. Although this assumption facilitates the inversion of the

variance-covariance matrix, it is not reasonable to assume that neighboring points located

in different tiles are not correlated. In Chapter 5 this assumption was relaxed and a random

partitioning model which takes into account the correlation between tiles was developed.

The model introduces at each locations as many random effects as the number of tiles and

therefore it requires the inversion of the spatial covariance matrix as many times as the

number of tiles. Future research may consider the approximation of tile-specific spatial

processes by sparse Gaussian processes (Seeger, 2003), as discussed earlier, estimating the

spatial processes from a subset of locations in each subregion. Inversion of the large,

original matrices would be replaced by the inversion of much smaller size matrices.

Models that account for non-stationarity allow mapping of spatial covariance parameters.

Estimates of these parameters are useful in improving the estimation of prediction error,

which helps quantifying the precision of the map. Low values of the range parameter

indicate that the spatial correlation decreases rapidly over short distances, suggesting that

the parasitaemia risk may be influenced by local factors such as human behavior, rather

than factors that vary over large areas.

In this thesis, the malaria maps of Mali and West Africa (Chapter 3 - Chapter 5) were based

on the analyzes of data extracted form the MARA database. This consists of published and

unpublished surveys which took place at different locations, including non-standardized age

groups of the study population. In addition, the season when the data were collected varied

between locations, making difficult seasonality adjustment of parasitaemia prevalence. Due

to age dependence of malaria prevalence, the maps were based on a subset of the MARA

database, which included malaria surveys carried out on a population with age ranging

from 1 to 10 years old. Gemperli et al. (2005) and Gemperli et al. (2006) have prove the

feasibility of using mathematical models of malaria transmission in converting heteroge-

neous malaria prevalence into a measure of transmission intensity, by employing the Garki


model. However, this model has been developed using field data from the savanna zone

of Nigeria and may not be reliable for other regions in Africa, with different environmen-

tal conditions and levels of malaria endemicity. In Chapter 6 a newly developed malaria

transmission model (Smith et al., 2006) was employed and maps of malaria transmission

and age-specific malaria risk maps for Mali were produced. The model is based on the

seasonal pattern of malaria by employing the seasonality model of Mabaso (2007). Model

validation revealed that the geostatistical model based on the estimates derived from the

transmission model had a better predictive ability compared to the one modeling the raw

prevalence data.

MARA is the most extensive database of historical malaria survey data in Africa. The data

collected since 1900’s contained in the database make MARA the perfect dataset for the

analyzes of temporal changes in malaria transmission. However, the maps based on these

data may not reflect the current situation of malaria because this could be influenced by

intervention measures that are not documented and therefore it is not possible to adjust for

them. Another major shortcoming of the MARA prevalence data is that they are collected

mainly in high endemic areas and therefore introduce selection bias. The necessity of

standard surveys and up to date, better quality malaria data led to the development

of Malaria Indicator Surveys (MIS), carried out in few African countries. The MIS are

nationally representative household surveys, running in a specific period of time. The

data corresponds to the current malaria situation and avoid sampling biases. In Chapter

6 parasitaemia data from the Zambia National MIS were analyzed and smooth maps of

malaria risk were obtained. To validate the use of mathematical model in malaria mapping

the parasitaemia risk map obtained by employing the transmission model was compared

with the risk map obtained directly by analyzing the prevalence data. Both maps predicted

similar patterns of malaria risk.

Throughout this thesis the importance of Bayesian geostatistical models to identify the re-

lation between environmental conditions and parasitaemia risk and to produce model-based

maps of malaria transmission was shown. These maps are essential for implementation of

malaria control programs because they offer valuable information on the areas which are

at high risk of malaria. Together with demographic data, the malaria distribution maps

may produce accurate estimates of the burden of disease and therefore optimize the allo-

cation of human and financial resources for malaria control. In addition, the malaria maps

provide a baseline against which the effectiveness of malaria intervention programs can be

127

assessed.

The country and regional malaria maps produced in this thesis can have an important role

in planning the intervention programs in these regions as long as they reach the key people

in malaria control programs, departments of health and research institutions in African

countries. Next steps toward the dissemination of this products include publication of this

work, distribution of poster sized malaria maps as well as organization of workshops in

collaboration with local partners.

Recently, there is a renew interest in malaria mapping (WHO, 2007), therefore new efforts

are put in collection of malaria data. Despite the above mentioned disadvantages of the

historical malaria survey data, we are currently updating MARA because it is one of the

only database to date which contains survey data across all Africa. The newly established

MIS are currently running in very few selected countries. We are planning to implement the

methodology developed in this thesis on the updated MARA database and together with

local collaborators to produce regional maps for East and South Africa and a continental

malaria map. The work in this thesis has shown that modeling assumptions make a

difference in malaria mapping and therefore it is important not only to collect data but

also to develop appropriate statistical models to obtain more accurate maps. This thesis,

following the work of Gemperli (2003), is an important step toward this direction.

Based on the results in this thesis the following suggestions can be made:

• Non-stationarity is an important characteristic of malaria that should be taken into

account when mapping the disease. In particular, the geostatistical non-stationary

model based on the fixed partitioning developed in Chapter 3 should be employed

when modeling malaria over regions with an apparent fixed partitioning. If the factors

that determine non-stationarity do not define obvious fixed subregions within the

study area, it is recommended the approach based on random Voronoi tessellations

developed in Chapter 5.

• The Bayesian P-splines approach is recommended for modeling non-linear effects of

environmental/climatic factors on malaria transmission.

• The mathematical transmission model employed in Chapter 6 can be incorporated

in malaria mapping to produce age and seasonality adjusted risk maps derived from

compiled survey data.

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Curriculum vitae

Laura GosoniuDate and place of birth: 27th May 1979 in Bucharest, RomaniaNationality: Romanian

Education

2004–2007 PhD studies in Epidemiology at the Swiss Tropical Institute, Baselon ”Development of Bayesian geostatistical modelsfor mapping malaria transmission in Africa”under the supervision of PD. Dr. P. Vounatsou

2001–2003 Master of Science (MSc) in Applied Statistics and Probabilitiesat Faculty of Mathematics, University of Bucharest.Thesis on ”ARIMA Series” under the supervision of Prof. Dr. M. Dumitrescu

1997–2001 Bachelor of Science (BSc) in Applied Statisticsat Faculty of Mathematics, University of Bucharest.Thesis on ”Multivariate Normal distribution”under the supervision of Prof. Dr. M. Dumitrescu

Professional Activities and Teaching

2001–2003 Statistician in the National Institute for Statistics, Bucharest2003 Statistics Lecturer in the College of Statistics, University of Bucharest2006–2007 Statistics Lecturer at the European Course in Tropical Epidemiology

Publications

Gosoniu L, Vounatsou P, Sogoba N, Maire N, Smith T. Mapping malaria risk in West Africa using aBayesian nonparametric non-stationary model. Computational Statistics and Data Analysis, Special Issueon SPATIAL STATISTICS, in press.

Gosoniu L, Vounatsou P, Tami A, Nathan R, Grundmann H, Lengeler C, 2008. Spatial effects of mosquitobednets on child mortality. BMC Public Health 8:356.

Sogoba N, Vounatsou P, Bagayoko M, Doumbia S, Dolo G, Gosoniu L, Traore S, Smith T, Toure Y, 2008.Spatial distribution of the chromosomal forms of anopheles gambiae in Mali. Malaria Journal 7:205.

Sogoba N, Vounatsou P, Bagayoko MM, Doumbia S, Dolo G, Gosoniu L, Traore SF, Toure YT, Smith T,2007. The spatial distribution of Anopheles gambiae sensu stricto and An. arabiensis (Diptera: Culicidae)in Mali. Geospatial Health 1(2), 213-222

Matthys B, Tschannen AB, Tian-Bi NT, Comoe H, Diabate S, Traore M, Vounatsou P, Raso G, GosoniuL, Tanner M, Cisse G, N’Goran EK, Utzinger J, 2007. Risk factors for Schistosoma mansoni and hookwormin urban farming communities in western Cte d’Ivoire. Tropical Medicine and International Health 12,709-723.

Gosoniu L, Vounatsou P, Sogoba N, Smith T, 2006. Bayesian modelling of geostatistical malaria riskdata. Geospatial Health 1, 127-139

Matthys B, Vounatsou P, Raso G, Tschannen AB, Becket EG, Gosoniu L, Ciss G, Tanner M, N’GoranEK, Utzinger J, 2006. Urban farming and malaria risk factors in a medium- sized town in Cote d’Ivoire.American Journal of Tropical Medicine and Hygiene 75, 1223-1231

Raso G, Vounatsou P, Gosoniu L, Tanner M, N’Goran EK, Utzinger J, 2005. Risk factors and spatial pat-terns of hookworm infection among school children in a rural area of Western Cote d’Ivoire. InternationalJournal for Parasitology 36, 201-210

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Development of Bayesian geostatistical models with ... · malaria epidemiology INAUGURALDISSERTATION zur ... Bayesian computation implemented via Markov chain Monte Carlo (MCMC) simulation